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Borel-measurable function

Prove that the following function is Borel-measurable function.

f_n(t) = { [t*2^n]*2^-n , 0 < t < n,

n , t > or = to n

| f_n(t) - t | < 2^-n , t < n }

I want a detailed proof. I want to know what one needs to check when proving some function is Borel-measurable function.

There are no typos, the way the function is defined for 0 < t < n is approximation for the function

Solution Preview

OK, now i think I understand: you have the notation [x] = "the integer part of x".

I shall use the following definition of a measurable function, applicable to any measure, including Borel measure:
A function f: X-->R is measurable if, for every real number r, the set
{ x in X: f(x) > r }
is measurable

If my understanding of notation [x] is not the same as yours or if the definition of measurable I use is not the one that you want to use, please DO NOT READ FURTHER ON, request a refund, and POST YOUR QUESTION AGAIN so that the management can see that you do NOT use the solution given ...