Welcome to lecture number 16 on measure and integration. If you recall in the previous lecture, we had started looking at the notion of measurable functions on measure spaces. We will continue that study of measurable functions and their properties and then we will specialize measurable functions on measure spaces. And then we will look at the space of measurable functions on when x is real line and the sigma algebras are the Borel sigma algebra or Lebesgue sigma algebra. And if there is a time, we will start looking at the integration of non-negative simple measurable functions. So, let us recall what we had been doing. We had been looking at properties of measurable functions. So, that is what we will continue doing. Then we will look at measurable functions on measure spaces, look at Borel and Lebesgue measurable functions and then look at integral of non-negative simple measurable functions. So, let us just recall what we had done regarding measurable functions. We had said that if x s is a measurable space and f is a function from x to r star, then saying that f is measurable, f measurable, it is same as saying the inverse image of every interval i belongs to s for every interval i contained in s. And there are equivalent ways of defining measurable functions in terms of special intervals like this is same as if and only if f inverse of the intervals of the type c to infinity belong to s for every c belonging to r and so on. So, and then we looked at what is called the algebra of measurable functions.  So, today we will look at the properties of sequences of measurable functions. So, we want to prove the following namely look at a sequence f n of measurable functions, then look at the function what is called the maximum of f n's. So, this is a function denoted by v n equal to 1 to infinity f n of x is defined as the maximum of f n of x n bigger than or equal to 1. So, this is called the maximum of the sequence f n and similarly we have the notion of a minimum of f n's which is noted by veg 1 to infinity f n x equal to minimum. So, this is extra, so that is the definition. So, claim is that if f n is a sequence of measurable functions, then the maximum and the minimum are also measurable functions. So, let us prove this. So, f n is defined on x to r star and f n is measurable for every function bigger than or equal to 1 and we define maximum n equal to 1 to infinity of f n of x to be equal to maximum of f n x n bigger than or equal to 1. So, this is the definition of this maximum and we want to prove to show that this function v n equal to 1 to infinity f n, this is measurable. So, to prove that, we can use any one of those conditions which we have defined earlier for measurable. So, let us look at maximum n equal to 1 to infinity f n of inverse of the interval, let us look at say c to infinity. So, that means what? That means this is all x belonging to x such that n equal to 1 to infinity f n of x is bigger than or equal to c. So, to prove this, now we have to convert somehow this relation into individual f n's because each individual f n is measurable. So, that seems, this is saying the maximum is bigger than or equal to c. That means at least one of them crosses over c. So, that is one way of doing it, but let us look at the equivalent criteria. Namely, let us look at the sets f n inverse, the maximum is less than or equal to c. So, that is equal to minus infinity to c. Let us look at that. This is same as x belonging to x such that the maximum value f n x is less than c. Now, if maximum of something is less than c, then each one of them has to be less than c. So, that is the reason instead of using this kind of intervals, it is more convenient for the maximum to use this kind of intervals because then this set can be written as intersection n equal to 1 to infinity of x belonging to x such that f n x is less than c. So, this may not be exactly true. So, let us take it this close because maximum less than or equal to c, then every one of them will be less than or equal to c. So, we look at the intervals of the type minus infinity c close and that is intersection of the sets and each one of the sets belong to the sigma algebra s because each f n is given to be measurable. So, it is the intersection of elements in the sigma algebra. So, this set also belongs to the sigma algebra s. So, that means this proves the fact that the maximum of f n is a measurable function. A similar proof will work for the minimum. So, let us look at the wedge n equal to 1 to infinity f n that is defined as the minimum of f n x for n bigger than or equal to 1. So, claim is that this is a measurable function. Once again for any c belonging to R, let us look at the minimum of f n n equal to 1 to infinity inverse of some type of intervals we want to show it belongs to s. So, let us try looking at minimum of this is bigger than c. So, let us try this. So, this is all x belonging to x says that the minimum of f n x n bigger than or equal to 1 is bigger than or equal to c. So, if the minimum of certain numbers is bigger than or equal to c then each one of them has to be bigger than or equal to c, because even if one is smaller then the minimum will become smaller. So, this is equal to intersection of n equal to 1 to infinity of all x belonging to x is that f n x is bigger than or equal to c. So, that implies and this is nothing but intersection n equal to 1 to infinity of f n inverse of c to plus infinity and each f n being measurable this is the measurable set. So, implies that this is a set in s. So, we have proved that if f n is a sequence of measurable functions then we look at the maximum or the minimum of the sequence of this measurable functions then that both of them are again measurable functions. So, as a consequence of this we prove that the limit of measurable functions is also a measurable function. So, that is our next aim to prove that the limit of measurable functions is also a measurable function. So, for that let us understand what is the limit of a function. So, let us look at a sequence f n the sequence f n of functions each f n is defined from x to star. So, to define the notion of limit of f n at a point x what we do is for every x belonging to x, let us look at the maximum or the supremum of f n x for n greater than or equal to some stage m. So, this number the supremum that depends on m and then we take the infimum over all m's. So, this is supremum first take the supremum and then take the infimum. This gives you a function this is called limit superior of f n at the point x. So, this is called the limit superior of the function sequence of functions f n x at the point x. Similarly limit inferior of f n x is limit inferior is defined as you first take the infimums of f n x for n greater than or equal to some stage m and then look at the supremum for all m bigger than or equal to 1. This is called the limit superior and you must have seen in your elementary analysis classes that limit superior of f n x is always bigger than or equal to limit inferior of f n x. So, this inequality always holds and the sequence f n f n x converges to f x if and only if limit superior f n x is equal to f of x. So, these are elementary effects from basic analysis about when is the sequence of real numbers convergent. So, it says that a sequence for any sequence of real numbers or extended real numbers you can define the concept of limit superior and also you can define the concept of limit inferior. Limit superior is defined by looking at the supremums of the sequence a n from some stage m onwards and then this supremum depends on m. So, look at the infimum of all these supremums. So, that is called the limit superior and similarly limit inferior is defined as first taking the infimums of the sequence f n x from some stage m onwards and then looking at the supremums of these numbers which depend on m. And one proves that the limit superior of a sequence is always bigger than or equal to limit inferior and the sequence is convergent if and only if the limit superior is equal to limit inferior. So, in case you have not come across these concepts strongly says that you pick up a book on elementary analysis and revise the concepts of limit superior and limit inferior. So, we are going to use that fact now here to prove that f is measurable. So, what is f? f of x is nothing but limit superior and limit inferior. So, only thing to show is that the limit superior and limit inferior are both measurable functions, but limit superior of f n is nothing but first taking the supremums of f n x from n bigger than or equal to m and then taking the infimums m bigger than or equal to 1. And just now we have shown that if f n is a sequence of functions then the supremums the maximums are also measurable functions. So, this is a measurable function, infimums of measurable functions that is a measurable function. So, this implies that limit superior f n is measurable and similarly limit inferior f n is measurable and saying that f n converges to f is same as saying this f is equal to limit superior f n or also equal to limit inferior of f n.  So, limits of measurable functions are also measurable functions. So, this proves the theorem that the class of all measurable functions is nice, it is closed under taking point wise limits. Now, let us observe that most of these properties hold for extended real valued functions also when properly defined.  But the problem comes the problem comes if f of x is equal to plus infinity and g of x is equal to minus infinity then what will be this number that is not defined or f of x is equal to minus infinity and g of x is equal to plus infinity even then the problem comes this number is not defined. So, what one does is to meaning suitably defined means look at all the points call this set as a where all x belonging to x where either of these things happen where f x equal to plus infinity g x is equal to minus infinity or f x equal to minus infinity and g x equal to plus infinity. Now, one observes that this set a is in the sigma algebra because f x is equal to plus infinity belong to sigma algebra intersection with that belongs to sigma algebra. So, all these sets all this set a is in the sigma algebra. So, a is the set on which the problem can come. So, what one does is one defines f plus g of x to be equal to f x plus g x if x does not belong to a and if x belongs to a you can define it any number alpha if x belongs to a. So, with this definition it is easy to observe. So, let me leave it as exercise for you to show that if I define it this way with alpha any value on the set a then f plus g is s measurable. So, that is what I mean by saying that the above results most of these properties hold for extended real valued functions also when those functions are appropriately defined. So, will not go much into detail of this one can easily verify these things. So, let us look at f and g to be another property of measurable functions is the following. Let us look at two functions f and g which are measurable. Then the following holds the look at the sets x belonging to x say that where f x is bigger than g x or the set x belonging to x where f x is strictly less than g x or x belonging to x where f x is equal to g x and similarly where f x is bigger than or equal to or f x is less than or equal to. So, all these type of sets claim is are in the sigma algebra s. So, let us look at to prove of one of them and others will follow similarly.  Let us look at the set x belonging to x such that f of x is strictly less than g of x. So, our aim is to show that this belongs to the sigma algebra s and since we are given f and g are both measurable, we are given the property that f x less than or equal to some real number is belongs to the sigma algebra and similarly g x less than or equal to a real number belong to the sigma algebra. So, objective is try to interpret this set in terms of union, intersections of something of sets of the type where f x is less than something and g x is less than something. So, for that we observe that for any x if f of x is less than g x then there must be a rational number in between them.  So, here we are using the effect that rational are dense on the real line. So, with this property we can write x belonging to x say that f x is less than g x. So, this implies that x belonging to x say that for some rational f x is less than r is less than g x for some and conversely if for some r this is true then obviously f x is equal to g x. So, claim is this is equal to union over r belonging to rational numbers. So, this is the only crucial point in this that the set f x less than g x can be written as a union over all rationals such that f x strictly less than r strictly less than g x. Now observe this set is an intersection. So, I can write it as r belonging to q this set is where f x is less than r and g x is bigger than r. So, it is x belonging to x say that f x less than r intersection with the set x belonging to x say that g x is bigger than r. So, what we have done? We have interpreted the set x belonging to x say that f x less than g x as let us just look at the set again. So, this is union over r that is equal to union over r belonging to q. What is this set?  The second set is the in nothing but g inverse of g x bigger than r. So, it is r plus infinity. So, this is so the set f x less than g x is written as union over rational intersections of these two sets. Now f and g being measurable this set belongs to the sigma algebra g being measurable this set belongs to the sigma algebra it is intersection. So, the whole set belongs to the sigma algebra intersection belongs to the sigma algebra and rationals are countable. So, this is a countable union of elements in the sigma algebra. So, this belongs to the sigma algebra. So, what we have shown is the set x belonging to x there f x strictly less than g x belongs to the sigma algebra s. So, that proves the first property of the theorem. Now, if we take just the complement of this set. So, also implies that x belonging to x such that f x less than g x the complement of this set what will be that. So, that is all x belonging to x such that f x is bigger than or equal to g x. So, that set also belongs to the sigma algebra and similarly the argument which will similarly we said that f x less than g x belongs to it.  So, f x bigger than so that will be r plus infinity in intersection with g inverse of minus infinity to r. By similar argument where we had f x less than again we can enter f x is bigger than g x. So, there must be a rational in between. So, that must be true and that will imply that this belongs to the sigma algebra. So, saying that f x bigger than g x belongs to the sigma algebra is and if you take the complement of this that is nothing but x belonging to x such that f x less than or equal to g x. So, that also belongs to the sigma algebra because this is the complement of the set in the sigma algebra.  So, these are nice properties and we will see use of these properties soon. So, with this we complete the study of measurable functions on measurable functions on measurable spaces. Next we are going to look at measurable functions which are defined on measure spaces. So, they play also a play role later on. So, we want to define we want to look at functions f which are defined on a set x taking extended real values and on x there is a sigma algebra. So, this is a sigma algebra S and a measure mu given on S. So, let us first define what is the meaning of the notion of almost everywhere. So, we say that a property p about the points of x is set to hold almost everywhere with respect to the measure mu. If you look at the set of points x for which the property p does not hold at x. So, look at all those points of x say that the property p does not hold at the point x. So, this is a subset and we want this subset belongs to the sigma algebra and mu of e is equal to 0. So, what we are saying is except for a set of measure 0 the property holds. So, that is why we give it a name that the property p holds almost everywhere with respect to the measure mu. Let me illustrate this with some examples. Let us take a function f and we look at the statement that f is 0 almost everywhere. So, f is a function which is extended function defined on the set x and we want to say that this function is 0 almost everywhere. So, look at the set of points where f is not 0. So, what will the statement mean? That set where f is non-zero should be element in the sigma algebra and its measure should be 0. So, x belonging to x say that f of x is not 0 that should be element in the sigma algebra and its measure of can be defined only when the set is in the sigma algebra. So, mu of that set is equal to 0. So, the statement that f is equal to 0 almost everywhere will mean mu of measure of the set of points where f x is not 0 is 0. Let us look at another illustration of this. The statement that f is finite almost everywhere what will that mean? So, that will mean look at the set of points where f is not finite that means what f is extended real value to function. So, it can take the value plus infinity or minus infinity. So, the set of points x belonging to x say that mod f x is equal to plus infinity. So, that is same as where either f of x is plus infinity or f of x is equal to minus infinity that set is in the sigma algebra and mu of that set is equal to 0. So, saying a function f is finite almost everywhere means the set of points where it can take the values plus infinity or minus infinity is a set of measure 0.  So, f is strictly bigger than g almost everywhere what will that statement mean? That means the set of points where f x is not strictly bigger than and that is the same as the set of points where f x is less than or equal to g x. So, the compliment of that statement is f x less than or equal to g x these set of points have got measure 0. So, saying that f x is strictly bigger than g almost everywhere with respect to mu means mu of the set where this statement is not true and that is f x less than or equal to g x is 0. This concept of almost everywhere is quite useful when looking at measurable functions. So, let us prove the property that if f and g are two extended evaluate functions say that f x is equal to g x almost everywhere mu then, measurability of one of the functions f implies the measurability of the other function g. So, f is measurable, s measurable if and only if g is s measurable. So, let us prove this property that if two functions are equal almost everywhere then the measurability is not changed one in measurability of one. So, let us look at f is from x to r star g is also from x to r star and we know that the set of points x belonging to x says that f x not equal to g x this set has mu measure equal to 0. So, let us suppose f is s measurable to show g is s measurable. So, to show that g is s measurable let us look at g inverse of any interval i. So, g inverse for every interval i i a interval in r star then g inverse of i we want to show that. So, claim is that this belongs to s for every interval i and now we have to transform this property this set into something regarding g. So, let us regarding f. So, let us look at g inverse of i is same as all x belonging to x such that g x belongs to i. So, this is a subset of the set x. So, what I can do? I can write this set as so g inverse of i I can write as intersection of so x belonging to x say that g x belongs to i and intersected with the set a also union. So, intersect with a complement. So, g inverse of i I have intersected with a and a complement. So, it is a union of these two sets x belonging to x say that g of x belongs to i intersection a complement. Now, let us look at the first set. So, this is g x belonging to y intersection a and what was the set a? What is the set a? Where a is the set x belonging to x where f x is not equal to g x f x. So, let us observe we are given that mu of a equal to 0. So, that automatically implies that a belongs to a belongs to the sigma algebra s and that automatically implies that the set x belonging to x say that g x belongs to i intersection a also belongs to the sigma algebra. So, this set also belongs to the sigma algebra. Why? Because this is a subset of a and a is a set of measure 0. So, this is a set of measure 0 and we have already assumed our measure spaces are complete. So, because of so the implies this is so because the measure space x s mu is complete. So, we have made the assumption that we are working on complete measure spaces. So, that gives that shows the importance of complete measure spaces. So, this set belongs to a and the other part a complement on a complement f is equal to g. So, I can replace it by a complement. So, the set x belonging to x say that g x belongs to i intersection a complement is same as the set x belonging to x where f x belongs to i intersection a complement because on a complement f is equal to g. So, that means what? So, g inverse of i is written as g inverse of i intersection a. So, let us just rewrite this statement again. So, what we are saying is g inverse of i can be written as g inverse of i intersection a union g inverse of i intersection a complement and that is same as g inverse of i intersection a union f inverse of i intersection a complement because on a complement f is same as g and this is a set of measure 0 mu of this set is equal to 0. So, implies that this set g inverse of i intersection a belongs to the sigma algebra and this set f is measurable. So, implies this set is in the sigma algebra a is in the sigma algebra. So, a complement in the sigma algebra intersection in the sigma algebra. So, this element belongs to the sigma algebra. So, and this is a union of two elements in the sigma algebra. So, this implies that g inverse of i also belongs to the sigma algebra s. So, we have shown f measurable f equal to g almost everywhere mu implies g measurable. So, that is the importance of measurable functions equal almost everywhere, but keep in mind we have used the fact that underlying measure space is a complete measure space. So, measurable functions that means this says that if f is measurable you can change its values on a set of mu measure 0 and still the function will remain measurable. So, another interpretation of this result is if f is measurable and you change its values on a set of measure 0 and call that function as g. So, that is measurable. So, that is a quite an important fact. Another application of this concept of almost everywhere is the following.  That is the set of points where f x is not equal to the limit has got this set, this set has got measure 0. Then the claim is then the f is also measurable. So, just now we proved that if a sequence f n of measurable functions converges to f then f is measurable and now we are saying that if f n are defined on a complete measure space and f n converges to f almost everywhere even then this property remains true. So, basically the idea is same as before. So, let us just look at how does one write a proof of this statement. So, we have got a complete measure space x s mu measure space.  So, that means look at the set A x belonging to x such that f n of x does not converge to f of x. Then what is given to us is that this set A belongs to the sigma algebra and mu of A is equal to 0. So, now let us look at, so we want to show that f is measurable to show f is measurable. So, once again look at for any interval i look at f inverse of i.  Now on A, so we are given A is a set of measure 0. So, this is a subset of A. So, that has set of measure 0. So, this implies that f inverse of i intersection A belongs to the sigma algebra S because this is a set of measure 0 and our underlying measure space is complete. On this portion A complement f n is converging to f. So, this f I can write as limit n going to infinity of f n inverse of i intersection A complement. So, this set is same as this.  So, that is a measurable set. So, this is a element in the sigma algebra because on A complement f n is converging. So, if we restrict ourselves to A complement then that must be element in the sigma algebra. So, both belong to sigma algebra. So, this belongs to the sigma algebra S. So, the concept of almost everywhere when dealing with the complete measure space is we can exploit that property. So, this implies that if f n is a sequence of measurable functions converging to a function f almost everywhere then the limit also is a measurable function. So, this emphasizes the property of something holding almost everywhere. Now, let us specialize the case when our underlying measure space is set is the real line. Then we have got two sigma algebras when x is equal to real line. Then we have got two sigma algebras one is the Borel sigma algebra and other is the sigma algebra of Lebesgue measurable sets. We have shown that the sigma algebra of Borel subsets is a subclass of Lebesgue measurable sets. So, when we are looking at functions defined on real line taking values as extended real numbers there are two possibilities to analyze whether the function is measurable with respect to the Borel sigma algebra or measurable with respect to the Lebesgue sigma algebra. So, there are two notions of measurable as far as real line is concerned and so we will separate them out. So, we will say a function is Lebesgue measurable if the inverse image of every interval in r star is a Lebesgue measurable set. So, if the inverse image of every interval in r star is an element in is a Lebesgue measurable set then we will say that function is Lebesgue measurable and we will say a function is Borel measurable if for every interval in r star we will pull back its image its pre image in r is a Borel set in r. So, here is a difference Lebesgue measurable requires that the inverse image is in the Lebesgue sigma algebra sigma algebra of Lebesgue measurable sets and f inverse of i in B r says it is the inverse image is everywhere is always a Borel set in r. So, obvious because Borel subsets form a subset of r. So, it obviously clear that every Borel measurable function is also a Lebesgue measurable function because inverse image of every interval if it is in B r and B r is a subset of l r. So, every Borel measurable function is also a Lebesgue measurable function. For example, let us look at a function which is continuous. So, if f r to r is continuous function then it is going to be a Borel function. So, let us prove that every continuous function is a Borel measurable function and hence also Lebesgue measurable. So, f is a function which is defined from x sorry x is real line. So, f is a function defined from r to r and f is continuous. Claim that f is Borel measurable that is f inverse of any set E belongs to B r for every set say E belonging to B r and continuity of a function can be expressed in terms of open sets. So, let us look at the class A of all subsets E belonging to B r such that this property is true f inverse of E belongs to B r. So, what you have to show? To show saying that f inverse of E belongs to B r for every E in B r it is an equivalent to saying to show that this A is equal to B r and that is where we are going to use our sigma algebra technique. So, to show that A is equal to B r note that A is already a subclass of B r because of we are picking up sets in B r. So, to show that A is equal to B r we have to show that B r is inside A. So, for that we will show two steps one open sets are contained in A and second will show that A is a sigma algebra because once A is a sigma algebra and includes open sets it must include the smallest sigma algebra generated by open sets that is B r. So, B r will be inside A and will be through. So, to prove these two facts are quite obvious because of the given condition so fun open sets belong to A.  So, what we have shown is if U is open then f inverse of U is in B r. So, that proves so implies that the open sets are inside A and A is a sigma algebra that is more set forward. So, let us observe empty set is equal to f inverse of empty set and so and r is equal to f inverse of r. So, both belong to A because empty set and the whole space they are equal to this. So, this is obvious empty set and the whole space belong. Second property if E belongs to A that implies f inverse of E belongs to B r and that implies f inverse of E complement belongs to B r and that implies because this set is same as f inverse of E complement that belongs to B r. So, what we have shown is if E belongs to A then f inverse of E complement belongs to B r and that implies that E complement belongs to A. So, that means E complement. So, the class A includes empty set includes the whole space it is closed under complements. So, finally, let us show that it is also closed under countable unions. So, that means let us take sets E n belong to A n bigger than or equal to 1. Then look at that means what we are given that f inverse of E n belongs to the sigma algebra B r because the property E n belongs to A means the inverse image is in B r. So, what that implies B r is a sigma algebra that implies union 1 to infinity f inverse of E n belongs to B r and now a simple observation that this set is same as f inverse of union E n n equal to 1 to infinity that belongs to B r. So, if E n belongs to A then f inverse of the union belongs to B r. So, that means union of n equal to 1 to infinity E n belongs to B r belongs to A. If E n belongs to A then f inverse of the union belongs to B r that means the union belongs to A. Hence, we have shown that A is a sigma algebra of subsets of A and it includes open sets. So, it must include B r and hence this is equal. So, that proves that every continuous function from r to r is Borel measurable and hence it is also Lebesgue measurable. So, all topological nice functions, continuous functions become Lebesgue measurable on the real line. Let us look at some more properties. So, we showed that every Borel function is Lebesgue measurable. So, there exist functions first of all r to r star which are not Lebesgue measurable. So, to prove that we can only simply observe that there are sets. Let us go back and recall that for a function f from r to say r star, let us look at f equal to indicator function of a set A where A is a subset of x. So, recall chi of A is Lebesgue measurable if and only if A belongs to L of r. So, if you can produce a set which is not Lebesgue measurable, then the indicator function will not be Lebesgue measurable. So, the answer to this question does that this non-Lebesgue measurable functions depends upon whether there is non-Lebesgue measurable sets. If you recall, we had proved the fact that the non-Lebesgue measurable sets exist. That question is related to basic set theory. So, if we assume exume of choice, then we constructed non-Lebesgue measurable sets. So, assuming exume of choice, one can claim that there exist functions which are not Lebesgue measurable. By the same reasoning, one can ask the question do there exist functions which are Borel measurable, but not which are Lebesgue measurable, but not Borel measurable because every Borel measurable is Lebesgue measurable. So, that means to for that by the same logic again, if we pick up a set A which is Lebesgue measurable, but not a Borel set, then the indicator function of that set is going to be a function which is going to be Lebesgue measurable, but not Borel measurable. So, these two questions that whether there exist non-Lebesgue measurable, non-Lebesgue measurable functions and whether there exist functions which are Lebesgue measurable, but not Borel get tied up with the fact that the Lebesgue measurable subsets is a proper subset of power set of R and B R is a proper subset of the Lebesgue measurable sets. So, with that we conclude the study of properties of measurable functions. So, basically let me just recall the measurable functions or functions defined on the underlying set X with properties that the inverse image of every set E in the Borel sigma algebra of extended real numbers. The inverse image is again in the sigma algebra on the domain space that is S. So, this is a property about the inverse images of sets being in the sigma algebra S and we will see that how this property plays a role in our further study of study of integration. So, we will do in the next lecture we will start the notion of integration for measurable functions. Thank you.


Ladies and gentlemen, the President of the United States, accompanied by his Eminence James Cardinal Hickey. Ladies and gentlemen, the President of the United States, it is my special honor and joy to present the President of the United States. Thank you very much. Thank you, Walden. Thank you, Cardinal Hickey. Now some of you may wonder why I've come here today. Well, I like great teams, and I couldn't think of two greater teams than the Lions and the Saints. I understand that you just had student body elections. You know, I have to tell you, I was President of the student body in my high school, and I always had a dream that one day the President of the United States might come visit our school. Of course, every time we invited him, President Washington said he was busy. By the way, a certain friend of mine has a message she wanted me to send to you. Please, for your families, for your friends, for your country, and for yourselves, just say no to drugs and alcohol. Cardinal Hickey, Bishop Carada, Father O'Malley, Sister Marcella Scully, Secretary Cavasos and Dan Curtin. Just before I came out here, my good friend Cardinal Hickey took me to view the altar before which Archbishop Carroll celebrated Mass. I was deeply honored that Cardinal Hickey gave me the opportunity to see where that great American man of God worshiped. America's first bishop was a scholar, a patriot, a good shepherd in our nation's founding years. St. Thomas Aquinas once wrote about qualities that marked the character of Archbishop Carroll. Three things, he said, are necessary for the salvation of man to know what he ought to believe, to know what he ought to desire and to know what he ought to do. I've come here today as a temporal leader, a man concerned with the affairs of state in the course of the country. And yet I've come to tell you, my young friends, that in all my years of public life, I have found that what Aquinas tells us is necessary for the salvation of man is also necessary for the strength and happiness of nations. Now we're in the middle of an election campaign, and everything I say is likely to be taken as political. But then even if I don't talk about where I stand, it's sort of like the story of a CIA agent who was sent to contact another agent in Ireland. He didn't know the other agent, but he was told that his name was Murphy and that to establish their contact, he was to say, it's a beautiful day today, but it'll be a greater day tomorrow. So he made his way to this tiny town by the sea and walked into the local pub, said to the bartender, I'm looking for a man named Murphy. And the bartender said, well, if it's Murphy, the bootmaker you want, he's across the street on the second floor. If it's Murphy the farmer, he's just a mile down the road and on the left-hand side. And then my name is Murphy. The agent said, well, it's a beautiful day today and it'll be a better day tomorrow. I said to the bartender, it's Murphy the spy you want. So you know where I stand even if I don't say so. But I hope you won't mind if for the most part I set aside the election. What I have to say to you today has to do not with the day-to-day politics, but with the enduring truths that mold men and women and move nations. Truths like faith, hope, and love, and as Paul tells us, the greatest of these is love. I found there are two kinds of people in this world, those absorbed in themselves and those who give love. Love to their families, to their friends, to their communities, to their country, and to God. Yes, we show love in many ways. By saying we love, of course, and by putting our arms around someone, but even more by how we live, by our courtesy, by our integrity, by studying and preparing for the future, and by service to humanity. Add it all up and you'd say by our values. Some in our age are inclined to say, well, that's okay, but not very important, so what else is new? But this is important and in many ways it's all so new. The American political philosopher James Q. Wilson has written that the most important change that he has seen since the mid-60s in scholarly thinking about how to make our country better is the new understanding, as he put it, public interest depends on private virtue. And to take just one area, education. Catholic schools across America are showing that private virtue and public interest do indeed live together. Yes, you have two of the best schools in this city and you're some of the best students in this city. And what's true of you, how you stand out is also true of the students of Catholic schools in most cities. Isn't that because you're not only learning the ABCs, but also about right and wrong, good and evil, and the nature of God's love.  The public interest in your education depends on the private virtues you're learning. Or as Aquinas might have said, it depends on your acquiring the elements of personal salvation. I don't want you to think I'm just talking here. I've heard a lot about your accomplishments and I couldn't help remembering something General George C. Marshall said when he was asked why he was so certain that we would win the Second World War. And General Marshall said we have a secret weapon, the best blankety-blank kids in the world. And when I was told about all you do, I thought America still has a secret weapon and it's still the best blankety-blank kids in the world. It amazes me that while you're exploring the mysteries of God's love and all that goes with it and showing how this exploration goes hand in hand with getting a good education, it's around our nation deny the public importance of the private virtues that you are mastering. If you can believe it, not long ago, one state chapter of a national activist organization said that for public schools to teach the idea that fidelity in marriage is a traditional American value would be unconstitutional since as they said, these values are rooted in religion. Well, God's love shows most strongly, of course, in the greatest gift of all, the gift of life. And here, as you know, there is great resistance to any talk about values. Recently, those who call themselves pro-choice have taken to discussing children who might be born deformed. Perhaps it would be better they say to spare the infant the struggle of life. I can't help thinking of Christy Nolan, who earlier this year received one of the world's most coveted literary awards. Why, Christy Nolan? Well, you see, there were complications at his birth that he almost died. And there were some who suggested that he should be allowed to, but doctors saved him. Only in the process, he was left totally paralyzed. He cannot walk, talk, or control his limbs.  In his message accepting the award, Christy Nolan wrote in that manner, Imagine what I would have missed if the doctors had not revived me on that September day long ago. Well, imagine what so many denied the right to life have missed. Imagine what we have all missed for their absence. Think of the cost to all of us because of the denial in public life of this most basic of values. I can't help wondering of those who call themselves pro-choice have ever stopped to think that the fetus, the unborn child, never has a choice. In no area is the importance of private virtue to the public interest clearer than in another area the area of drug abuse. When we came into office eight years ago we found a drug epidemic that few in the government seemed to care much about. We started arresting and sending to jail drug dealers and drug kingpins in larger and larger numbers. In the last eight years federal narcotics convictions have more than doubled and we have seen tons of cocaine and tens of thousands of tons of marijuana and there's other good news too. After much prodding, pushing and bludgeoning from us our reluctant Congress is expected to pass a tough new drug bill in the next few days. It would give our law enforcement officers new and better tools for helping them protect us all and to help protect the lives of the innocent. It would provide for the death penalty for those who commit murder in the course of a drug related crime. In the last eight years federal drug dealers and drug dealers and federal friends are dropping their nostalgia for the do your own thing in your own time baby, 60's and are joining us wholeheartedly in this fight against drugs. But important as all this government activity has been for my money the turning point in the fight against illegal drugs came when a certain little lady opened her heart and spoke the courage of the values that God placed in your soul. I've told this story before Nancy was in Oakland, California some time back speaking in a school room and a little girl stood up and said but what do we do when someone offers us drugs and Nancy said just say no. Today there are over 12,000 just say no clubs in schools across the country. The, you know something while the number of drug users soared during the four years before we took office it's dropping now and earlier this year we got the best news of all. High school students are saying no to drugs including cocaine as never before. What Nancy has been saying to so many young people is what the priests the nuns and the teachers say to you each day that you must have values to guide your lives. Too often values aren't taught or can't be taught in our public schools but they are taught here and may I say because you're here each of you is greatly privileged. But with each gift goes an obligation and yours is to act as examples to your friends who aren't as fortunate to go to these schools and who may be tempted by those who would lead them astray. I know that your parents all make great sacrifices so that you can come here it's a measure of their love for you. For years I've been urging congress to recognize the public interest in your education and to allow your parents to support your education either through tuition tax credits or vouchers. We need a congress that shares and supports the values of the American people. We hear the cry but what values do you mean? Well that's easy just for starters love thy neighbor as thyself in this past Lenten season the holy father invited Cardinal Hickey to give the yearly retreat for him and his household an honor never before accorded to an American priest or bishop. In his meditations the Cardinal said to obey God the author of our freedom is to respect our freedom and he added in the logic of the gospel harmony with God's will is the true definition of history. So this is my message to you as a secular leader but also as a man standing in humility before God to seek what the Cardinal calls true freedom to reach for what Aquinas called the necessities of salvation for if you do if these lessons become part of the instruction you carry with you to end your studies here America will be stronger the world will be better and there will be no limits to what in this sweet land of liberty you can do with your lives. Let me just if I can say a few words on my own about this this nation of ours you know I received a letter we're quite unique I received a letter from a man one day he pointed out something I'd never thought of he said you can go to live in France you cannot become a Frenchman you can go to live in Germany or Spain and you can become a German or a Spaniard and he went on naming Japan and China and other countries but he said anyone from any corner of the world can come to America and become an American and this this country is the only one you can say has that peculiar melding of people together revealing as no other area ever has that we are all the sons and daughters of God you know I don't say this very often and sometimes people may call it mysticism but I have always believed that there was some divine power and plan that placed this great continent between the two great oceans to be found only by people who had that extra love of freedom and that courage within their hearts to uproot themselves from their native land leave and sometimes family and friends but to come here and to create this nation that we have created for ourselves here I have to believe that that is true just as I believe that Lincoln spoke the truth I've learned in these eight years as never before when he said I could not perform the functions of this office if I did not know that I could call upon one who is stronger and wiser than all others thank you all God bless you all my name is Sonia Sweetney and I'm the student council president for All Saints High School on behalf of all the students here I would like to thank you for that inspirational speech and it is my pleasure to present to you from all the students and faculty of All Saints High School a yearbook and two t-shirts for both you and Mrs. Reagan God bless you and thank you for the motivating message good morning Mr. President my name is Jermaine Teller it is an honor for me to present to you this jacket with the number I think you will remember from your playing days on behalf of the 1987 Metro Conference Football Champions the lead in divisional play the Archbishop Curle Alliance


Welcome to Adventures in Small Business, a collaborative effort by the U.S. Small Business Administration Hawaii District Office, Hawaii Small Business Development Center, the Patsy Teeming Center for Business and Leadership, and the Veterans Business Outreach Center of the Pacific to showcase the stories of local entrepreneurs and small businesses. I'm Colleen McLooney from the Patsy Teeming Center for Business and Leadership, and today we have Brian and Anjel Foster from Island Olive Oil Company. Welcome. Thanks for having us, Colleen. Thank you so much for being here. So, let's see, today we want to start out with, well, I know, why don't we just first start with where your stores are located so then that'll give everybody an idea of what we're talking about. Okay. Well, currently we have two stores. We have the one that's at the Ward Center. It's in the Ward Village area. It's real easy and convenient. And then we also have another store that's in Kailua, it's in the Kailua Town Center, right on Kailua Road between Miochi-Ram and in California. Oh, fantastic. So, the Ward location was your first. Yes. How long have you been there? Seven years. Oh, seven years. Fantastic. And then how long have you been in Kailua? Just over two years now. Oh, okay. Oh, wonderful. That's so exciting. Oh, that's a big leap, too, for our small business to go from one location to two. Yes. That's it. Congratulations. Thank you. A little stressful at first, but after the dust settles and you start to see the bits of your labor, you kind of get a little more ease. You're like, okay, thank you. Right. Right. Oh, but that's fantastic. Congratulations. Thank you. Okay, so why don't we start out with sharing your background and about yourselves and how you got started with the Olive Oil Company? Interesting story. We both have very diverse backgrounds. We both grew up locally. Then we both went to school in the mainland. She was involved with fine arts, design, things of that nature. I was in finance and I went, 9-11 happened. I joined the military for a while, got into operations, went into defense. I was overseas a lot for what I did for work and she was more or less here most of the time. She was my rock and it was great because we were ready for a change and we talked about it and it was time for me to be more stable and be at home more often and we kind of just were talking and developing new ideas and new things. The Olive Oil Company, it was a match made in heaven because we're both really, really big into food. We've always been big nutritional people and everything else. We took a chance and that's what entrepreneurs do. We all have to take that leap of faith. We said, you know, there's no turning back now. Once we closed this door of very successful careers that we were in and transitioned into something new, it's a little stressful. But we did it and we haven't looked back and we're fortunate and blessed. Well it wasn't completely new because we both had a lot of background growing up in different food industry jobs. Ryan worked for several years putting himself through college and when I was young, my mom had a restaurant and I have lots of family members that have had restaurants. So we spent a lot of time working in food in the back of the house and all the different aspects and learning about small businesses and how to run them. So that was kind of... That's extremely helpful. Yes, yes. Oh, fantastic. Okay, so you decided to buy the business and or start the business and then let's see. So, okay, so your motivation was that you were ready for a change and that you wanted to take a new... Spend more time at home. Spend more time at home. In Hawaii. Correct. Excellent. Excellent. It's always so interesting to hear what motivates entrepreneurs to start their own business. It really is.  Early midlife crisis. Yes. Good motivation. Need a change. I was always on the road. Oh, you know. Right. It was just something we could do together was, you know, we both decided to bring our different... It was good because we could both bring our different strengths, the creative side and the operational side. Right. It's a match made in heaven. So can you share with us how maybe how hard it is being an entrepreneur? I know there's good sides too, but a lot of people, it's easier to share the harder side sometimes first. But it always is. I'll start with this. If it was easy, everybody would do it. You know, when you work for, I worked for a Fortune 500 company and when you're working for somebody else, you feel that motivation, but you're going in every day, you're getting a paycheck. You know, you can go home at night. Not a lot of stress. It depends on the position you're in, of course, you know, digress. But when you're an entrepreneur, not only do you have to worry about all your employees, you have to worry about yourself, you have to worry about all these different things and factors. You have to worry so many different hats.  And you're like one of those circus performers. You've had, you're juggling all these different balls. Nobody just keeps throwing more and more balls at you. Oh my God. And it never stops. And it never stops. It's 24-7. Yes. You know, but... 25-7 sometimes. Yes. Yes. But that's the stressful entity of it. You know, people, like you said, people want to talk about the negativity aspect of it. But the positive aspect of it is the reward that's seen people and the smile on their face. You know, we have some of our customers that come in and they're so thankful that we have these products that we have.  We can kind of make our own schedule. You know, we technically work 24-7, but we can, it's like, okay, if I really need a day off to go to the beach or something or we can do that, but you have to weigh them. It's very, very rewarding. It doesn't always feel like work when it's work, even though it's a different feeling. I think you have a little bit more batteries in you when you're, you know, working for yourself. Right. Got a lot of skin in the game. Yeah. Exactly. Oh, yes you do. Okay, so tell, let's see, tell me a little bit about some of the challenges you've encountered becoming entrepreneurs and being entrepreneurs and how you've handled them or the outcome for them. I'd say a lot of the other business owners that I've talked to, we all face the same things. The biggest one is labor, you know, getting the right staff in place. Hawaii is, of course, for such a low unemployment rate and getting good, qualified candidates in the door to help, because if, you know, as an entrepreneur, if nobody's there to work that shift or nobody's there to do that job, if it falls on us, exactly, you know, we're the ones doing it. So labor is, is the big, the big portion of things. Absolutely.  You know, as soon as you have someone trained and they're with you for a little while and then, and they can get hired away. Yeah, right. That's a challenge. Another challenge that we faced was the educational aspect of olive oil. A lot of people think that, you know, what they've been consuming their entire life is quality stuff. And so we have to educate all the loved customers that come in. We educate the chefs. We go and do teaching classes so that people are informed about what you're looking for in the quality oil, how that relates back to them, the difference between a fresh oil and a fresh versus a non not so fresh, I should say not so fresh oil. Right. The difference is night and day. Like the tickle in your throat. Yeah. Absolutely. So we're constantly educating and reinforcing that with our customers. You have a great amount of education then for your employees and too for your staff. Yeah. Absolutely. Oh, amazing.  You just said it. Program? The initial, the initial startup phase is we have, we give them a book of knowledge. And that's what it's called, island olive oil book of knowledge. It breaks down all the differences about olive oils, balsamic vinegar, scorn may products, water truffles. We go into this whole thing of it. So when we bring on somebody new, we're training them for an entire month and then after that month is up, then we kind of let them, let their wings let them fly a little more economy to them. And then, but we're, they're constantly learning as well. Because there may be things that come up in a conversation with a customer that we've never thought of before.  Yeah. It's always a new question. There's always a new question. Because somebody read something somewhere, somebody saw something. New recipes. Yeah. And you know, so we're constantly evolving and transforming. But we do train them. We do work with them for probably a month, at least that they go with another, another staff member and mainly Brian, the real trainer. That's an incredible amount of time. It's a lot of time to invest. Right. Amazing. A lot of labor hours. Yeah. Yeah. Yeah. Yes. For sure. Oh. It's wonderful to know though that your staff is so well educated. That's one of the things people, we, one of the biggest comments that we get is the customer service and that is the experience and customer service is what people look for. And nowadays there's so many places to spend your dollars that having those good experiences really matter. Well, it makes it, it really helps when you know that you're going to a retail store that has so much, they have so much knowledge about their product and then it makes you want to part with your dollar more because you realize the value of it. That's fantastic. So, okay, let's see. What about the competition that you have at Ward and Kailua for the products that you're carrying? How do I put this nicely? No, we don't really have, I'm sorry. No, no, no, we really don't have any direct competition. We have indirect competition with grocery stores and things of that for our dollars. But in terms of the freshness and quality, we have everything that's lab tested. Nobody else can tell that. We also have the freshest quality oils. A lot of people, we go the extra mile to make sure that we're the creme de la creme of all of Olavol's and Balsamacanaga. So people that know and they've gotten into it, they keep constantly coming back to it. So we've got a great recurring business that happens. People that refer us, the refer doctors, I don't know where these guys come in. They refer their patients to us and say, if you want to quality oil, go here. Because they take it as a nutraceutical, so it's very good for antioxidants and anti-inflammatory properties. So they're not only cooking with it, they're actually taking like a shot of olive oil. We do have a number of customers that use it as a nutraceutical supplement. Okay. So are there specific oils for that or is it more their choice of flavor when they go into the store? I mean, do you have a specific oil for that just to drink? That's an excellent question actually. So the fresher an oil is, the higher, it's called polyphenols, which is the antioxidants, anti-inflammatory, so it's good healthy properties. So as an olive oil, as soon as it's crushed, it starts getting older. So olive oil is only good for up to two years from the time it's actually crushed. So with that being said, think of it, you crush an olive and it's on a downward slope as soon as it's going, right? So it doesn't matter about when you open the bottle. It starts going. So since we're always rotating and we're bringing in the freshest, highest quality stuff, doctors will be like, hey, go there, you get that stuff. So we have it in the stores labeled out from mild to robust. And once we have it lab tested, we know what the phenolic content is within that oil. So a lot of these people come in and say, well, where's the high polyphenols? And they know. They're like to the right. So it's the right. They're like, OK. They don't care what it tastes like, although it is delicious. But they just want the one that has the highest polyphenols. So generally speaking, a commercial oil will probably have under 100 polyphenols. And our highest polyphenol oil we've ever had has been a general. Over 1200. OK. So that's a lot of anti-accidents. And what is the doctor usually recommending that they would want a polyphenol? High polyphenol. I don't know what. I am not a medical professional. Oh, yeah. No, I understand.  No, no. They just, they come in and they say, my doctor told me to come here because I need high polyphenols. Say this is the one. That's high. I don't know if he specifically recommends a certain number. I don't know. But it's a great referral and recommendation source. And that's a wonderful addition for your business, too, is the nutritional side, too. Absolutely. That's fantastic. My biggest recommendation with that is you can get that nutraceutical supplement in just making a salad and cooking with your high quality olive oil. Right, right. So, you know, it's great to have it that way, too. Great. OK, well, we're going to take a break right now and we will be back in one minute. Thank you. Hello, I'm Marcia Joyner, inviting you to join us on Wednesdays at 1 o'clock for the Cannabis Chronicle, the 10,000-year odyssey where we take a look at cannabis as food, cannabis as medicine, cannabis and religion, cannabis, and dear old Uncle Sam. So, please join us to learn all about cannabis again Wednesdays at 1 o'clock. Thank you. Hi, guys. I'm your host, Lillian Cumick, from Lillian's Vegan World. I come to you live every second Friday from 3 p.m. and this is the show where I talk about the plant-based lifestyle and veganism. So we go through recipes, some upcoming events, information about health, regarding your health, and just some ideas on how you can have a better lifestyle, eat healthier, and have fun at the same time. So, do join me. I look forward to seeing you and hello, Hal. Welcome back to Adventures in Small Business. We're here today with Anjel and Brian Foster from Island Olive Oil Company. We just had a great conversation about the nutritional value of olive oil, which is fantastic. And let's see. We are going to show some photos, I think, right now of the product, and we can talk about that, and then a picture of you in your new store, I believe. Is it the Kailua store? Yeah. Let's take a look at the pictures. Okay. So what are we looking at here? Right here, we have our Mediterranean citrus dipping blend. So a lot of people love to have herbs with their olive oil so they can dip bread, but we also use these for general cooking. You can add a little bit to salad dressing, to mix a salad dressing up. Use it to saute and roast veggies or with chicken. I use it for making soups, as well as our brownie kits and our popcorn kits. So we do a lot of little take and bake kind of goodies, and you can use it as a salad. That sounds wonderful. That's great. Okay. And then let's see our next picture. Ooh, what's that? These are shrubs. So these are our white balsamic vinegars, and they're already infused with fruit flavors, which is what a shrub is. So you can use them as a refreshing drink mixer. And right here, we've got a cranberry pear white balsamic with, and you can add a beautiful ruby red color, and just a little seltzer water and some mint. And there you go. You've got an alcohol optional beverage. Oh, that's fantastic. That looks wonderful, especially right now. Yes, exactly. We're doing this hot summer we're having. Very thirst-quenching and probiotic. Ooh, terrific. Okay. And then our last photo. Great. Is this the new store? That is. That is the Kailua store. I'm not sure this timeframe, when this will be taken. That's right around when we did an opening party with the Kailua Chamber of Commerce. That's right. That's right. Yeah, I remember that. Okay. And so there you can see the olive oil to the right of Brian there. You can see, is that olive oil there? That is actually white balsamic vinegar to the right. Oh, okay. If we're actually looking, the direction that we're actually looking would be towards the olive oils, towards our special tools like our truffle oils and things of that nature. But we have over 50 different varieties, so you can taste everything, and that's kind of how the store is set up so everyone can taste it. So you can see the little taster bottles on the side. Right. You can come up, grab some cups, and have some fun and enjoy. Oh, that's fantastic. Taste the difference. You can try almost everything, at least where the olive oil and the vinegars and all the gourmet products, anything we've got, like honeys and jams and topenades and salts and all kinds of things. It helps you to understand and know what you're actually getting. And it's part of the experience, and it helps create that, you know, really, and thoughtfulness in all of our customers get really enthusiastic about trying them. Right. And then as far as recipes for customers, I know that you have some on Instagram that you post pretty regularly. Yes, we do. And we're also doing, we have a lot on our website. We're always putting them on our Facebook page. And we're actually, we just had, we're doing a lot of recipe development right now. Oh. Great. And are you guys the recipe developers? Are you guys test all of your own recipes? Yes. Mostly. I get to test them. She creates them and I get to test them. We just, we just did an olive oil cake. We just did a, we just did a smoked olive oil, olive oil hummus. So we've got lots of really diverse, yummy kinds of fun things. That's so good. One goat cheese salad, so all different kinds of areas. That's also, it's not just her that's doing it or myself. You'd be surprised because you, you build that social on Instagram and people will actually post some recipes that they've done before and it just, it's built that momentum and we'll share it. It's a real community. Yes. Community effort. Oh, that's fantastic. Okay. I'm going to have to check that more often. Definitely. Okay, great. So let's talk about maybe some lessons that you've learned as entrepreneurs or insights maybe that you could share with other small businesses. Well, one thing I will say from a retailer is location, location, location, location. Location is everything. You want to think about the development in the area that you're doing your lease. You want to think about, you know, what the future plans are for the, that community and if your business fits that environment, you want to think about what kind of neighbors you have and if their customer base is complimentary to yours. So if you have neighbors that have shops or restaurants that have a similar clientele, that's a really great fit for you and you'll get a lot of crossover business and can do cross collaborations. Also, you want to think about foot traffic, right? You want to think about our, and parking. Parking is a big thing here in Hawaii. It's amazing, but you know, people really need to have that ease of parking. That's a big issue, you know, so location. Just really investigate your location. Fantastic. Have you got another insight, maybe, or? No, before we, before we, I was just going to touch on, based on that, before we opened up the Kailua location, we were actually going out there several times a week, different times of the day, just kind of taking a look at the foot traffic, seeing what was out there, sitting and watching, getting an idea and taking it all in, that kind of helped our decision making process. Whether or not you want to be freestanding or do you want to be in a mall. I mean, those are like all really good questions to ask yourself for a small business, running out. Definitely. Definitely. Oh, thank you. Thank you for sharing that with us. So, let's see. What is your vision of success then for Island Olive Oil Company? Is it changing all the time? Is there one big goal you're looking to achieve? Lots of things, yeah. I would say two things that really stand out to me, more or less, it defines success for us. One, is not worrying about the business when you go home at night, because a lot of entrepreneurs do still sit there and they have things, and if we can minimize that to a certain level of degree, I think it gives you a lot of peace of mind. That's part of it. And I think the other part is knowing that you made an impact in the community. When we're at the gym working out, you know, I remember one time I'm walking down the stairs and just another patron at the gym was walking up the stairs.  You're the olive oil guy.  It's like, I've got your stuff on my counter at the house. And that is like, you know, it's almost a huge connection. Yeah, it's a huge phone. You didn't even know about it. Exactly. And it's just, it's very gratifying and very, you know, fulfilling. It's like, wow, okay, they've got, we hear these stories all the time. Hey, I've got, I love your stuff. It's so awesome. You know, I've got it as a gift, or I've got it at that. That kind of, that's our incentive and that's our drive. Makes you feel really good. You know, it's just a sign of success when people have your product. When people recognize it. Yeah. Exactly. When people recognize your brand. That is pretty good. Definitely. We do appreciate that. And they recognize you specifically, too. Very hands-on. Very hands-on with the business, and I think that's what you have to be to be an entrepreneur. Well, customers really appreciate that. They really do. They do. Yeah. Okay. So our next question then is where do you see yourselves in about five years? We won't go to ten. Constantly evolving. Lake Homo. Yeah. That might be the ten-year mark. Okay. I mean, probably a little. Yeah. Maybe twenty. I don't know. But, you know, the business is constantly evolving, it's constantly changing and morphing with the times, right? So, we eventually want to go east, and we want to take west from Hawaii. We've got other locations to look at. We can look at licensing in Japan. Asia Pacific. Asia Pacific, Australia, things of that nature. So, some exporting, maybe. Correct. I can't. In fact, there's a class that you guys are doing on exporting. Right. Thank you. Awesome. And I'm actually going to be attending that class, because it's going to give me some insights. And I'd love the classes that you guys put on. What was the last class that you did with the Google Analytics? Google Analytics. There's such a resource for everyone that we can always get involved with. We're always learning. But that's kind of where we want to be, licensing. That's very exciting. And more e-commerce, I'd say too. Maybe two stores. One, definitely. We've already got that in the works of where we want it to be. We're just waiting for things to finalize.  Right. I can't wait to personally know if there's going to be another store in the works that's going to come in here. Oh, that's very exciting news. Yay. I can't wait to find out about that. The new baby. Yes, that's right. That's right. And then I think you have a collaboration coming up soon with Alan Wong. Yes, we do. We're really excited about it. Chef Alan Wong and his team have been huge supporters of our business, as well as wonderfully gracious customers and user products. What they are doing is called a omokase drink. Omokase. And they're using our balsamic vinegars as shrubs to create a little flight of cocktails and delicious, delicious nauseous, delicious nauseous to enjoy. Oh, that sounds very exciting. Very exciting. Very exciting. Oh, good. And that's coming up within the next couple of weeks? Yes, it's coming up next week. Oh, okay. And they do it for a limited time. It's like a special run. Oh, okay. It's not just a one night event. No, it's going to be for a week. So you can call and make reservations or go on Alan Wong's website. Oh, okay. Oh, that's wonderful. That sounds terrific. All righty, great. And then, let's see, I just want to mention one other thing that you were a part of was the KCC Culinary Fundraiser. Yes. Great point. Thank you. The KCC Culinary Team, popular in the community college, of course. We've been supporters of them for a few years now. And they, it's the national championships. Again, we just got word the other day that it was completed, that they got the silver medal. So they got second place. Right. We did a huge event outside, inside the ward store in that area. And we raised up a few thousand dollars from the team. Very excited about that. We'd like to help promote them. We kind of helped them give back to the community as well. You know, the community has been great to us. So we need to reciprocate and give back to others as well. Oh, that's fantastic. And support young chefs. Right. That's our business. Education. So innovative. Innovative in what they do is incredible. They sell minds. It is. It is. It's amazing. It's amazing. Well, we're going to wrap up. I just wanted to remind everybody about the Shop Small Hawaii Initiative that we have at the Pati Tuming Center, Business and Leadership, along with the SBA partners. So if you are a small business, please don't forget to register on Shop Small Hawaii. Island Olive Oil is going to become a neighborhood champion in Kailua. And if you have any questions or need any information about that, please don't hesitate to contact us at mpblhawaii.org or shopsmallhawaii.com. And we do have that exporting workshop coming up on the 28th of this month. And that will be at our YWCA Kokokahi location. And it will be from 11 to 1 at the Brown Bag Lunch Seminar or workshop with Mary Dale from the SBA district. All right. I want to thank Brian and Anjale Foster from Island Olive Oil Company today for joining us on Adventures in Small Business. And I hope to see you again next month. Thank you.