---
abstract: 'In this paper we invastigate the notion of generalized $(\ci,\cj)$ - Luzin set. This notion generalize the standard notion of Luzin set and Sierpi[ń]{}ski set. We find set theoretical conditions which imply the existence of generalized $(\ci, \cj)$ - Luzin set. We show how to construct large family of pairwise non-equivalent $(\ci,\cj)$ - Luzin sets. We find a class of forcings which preserves the property of being $(\ci,\cj)$ - Luzin set.'
address: ' Institute of Mathematics and Computer Science, Wroc[ł]{}aw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroc[ł]{}aw, Poland.'
author:
- 'Robert Ra[ł]{}owski and Szymon Żeberski'
title: Generalized Luzin sets
---

Notation and Terminology
========================

We will use standard set-theoretic notation following [@jech]. In particular for any set $X$ and any cardinal $\kappa$, $[X]^{<\kappa}$ denotes the set of all subsets of $X$ with size less than $\kappa.$ Similarly, $[X]^\kappa$ denotes the family of subsets of $X$ of size $\kappa.$ By $\cp(X)$ we denote the power set of $X$.

If $A\subseteq X\times Y$ then for $x\in X$ and $y\in Y$ we put $$A_x=\{y\in Y:\; (x,y)\in A\},$$ $$A^y=\{x\in X:\; (x,y)\in A\}.$$ By $A\vartriangle B$ we denote the symmetric difference of sets $A$ and $B,$ i.e. $$A\vartriangle B= (A\setminus B)\cup (B\setminus A).$$ In this paper $\cx$ denotes uncountable Polish space. By $\Open(\cx)$ we denote the topology of $\cx.$ By $\Borel(\cx)$ we denote the $\sigma$-field of all Borel sets. Let us recall that each Borel set can be coded by a function from $\omega^\omega.$ Precise definition of such coding can be found in [@kechris]. If $x\in\omega^\omega$ is a Borel code then by $\#x$ we denote the Borel set coded by $x.$

$\ci$, $\cj$ are $\sigma $-ideals on $\cx,$ i.e. $\ci, \cj\subseteq\cp(\cx)$ are closed under countable unions and subsets. Additionally we assume that $[\cx]^\omega\subseteq\ci, \cj.$ Moreover $\ci, \cj$ have Borel base i.e each set from the ideal can be covered by a Borel set from the ideal. Standard examples of such ideals are the ideal $\bbl$ of Lebesgue measure zero sets and the ideal $\bbk$ of meager sets of Polish space.

Let $M\subseteq N$ be standard transitive models of ZF.\
Coding Borel sets from the ideal $I$ is absolute iff $$(\forall x\in M\cap \omega^\omega)( M\models \#x\in I\iff N\models \#x\in I).$$

We say that $\ci$ satisfies $\kappa$ chain condition ($\kappa$-c.c.) if every family $\ca$ of Borel subsets of $\cx$ satisfying the following conditions:

1.  $(\forall A\in\ca)(A\notin\ci)$

2.  $(\forall A,B\in\ca)(A\neq B\rightarrow A\cap B\in\ci)$

has size smaller than $\kappa.$ If $\ci$ is $\omega_1$-c.c. then we say that $\ci$ is c.c.c.

Let us recall that a function $f:\cx\rightarrow\cx $ is $\ci$-measurable if the preimage of every open subset of $\cx$ is $\ci$-measurable i.e belongs to the $\sigma $-field generated by Borel sets and the ideal $\ci.$ In other words $f$ is $\ci$-measurable iff $$(\forall U\in \Open(\cx))(\exists B\in \Borel(\cx))(\exists I\in\ci)(f^{-1}[U]=B\vartriangle I).$$

Let us recall the following cardinal coefficients:

$$\begin{array}{l@{\ =\,{} }l}
non(\ci) &  \min\{ |A|:\; A\subseteq \cx\land A\notin \ci\}\\
add(\ci) &  \min\{ |\ca|:\;\ca\subseteq \ci\land \bigcup \ca\notin \ci \}\\
cov(\ci) &  \min\{ |\ca|:\;\ca\subseteq \ci\land \bigcup \ca=\cx \}\\
cov_h(\ci) &  \min\{ |\ca|:\;\ca\subseteq \ci\land (\exists B\in \Borel(\cx)\setminus\ci)( B\subseteq  
       \bigcup \ca) \}\\
cof(\ci) & \min\{ |\ca|:\; \ca\subseteq \ci\land \ca\text{ is a base of }\ci\}
\end{array}$$ where $\ca\text{ is a base of }\ci$ iff $\ca\subseteq\ci\land (\forall I\in\ci)(\exists A\in\ca)( I\subseteq A).$

Let us remark that above coefficients can be defined for larger class of families (not only ideals).

We say that $L\subseteq \cx$ is a $(\ci,\cj)$ - Luzin set if

-   $L\notin \ci,$

-   $(\forall B\in \ci)( B\cap L\in \cj).$

Assume that $\kappa $ is a cardinal number. We say that $L\subseteq \cx$ is a $(\kappa,\ci,\cj)$ - Luzin set iff $L$ is a $(\ci,\cj)$ - Luzin set and $|L|=\kappa$.

The above definition generalizes the standard notion of Luzin and Sierpi[ń]{}ski sets. Namely, $L$ is Luzin set iff $L$ is generalized $(\bbl,[\bbr]^{\le\omega})$ - Luzin set and $S$ is Sierpi[ń]{}ski set iff $S$ is generalized $(\bbk,[\bbr]^{\le\omega})$ - Luzin set. The above notion generalizes also notions from [@C].

We say that ideals $\ci$ and $\cj$ are orthogonal if $$\exists A\in\cp(\cx)\; A\in I\land A^c\in \cj.$$ In such case we write $\ci\perp \cj$.

Let $\cf\subseteq \cx^\cx$ be a family of functions. We say that $A,B\subseteq \cx$ are equivalent with respect to $\cf$ if $$(\exists f\in \cf)\; (B=f[A]\vee A=f[B])$$

We say that $A,B\subseteq \cx$ are Borel equivalent if $A, B$ are equivalent with respect to the family of all Borel functions.

We say that $\ci$ has Fubini property iff for every Borel set $A\subseteq\cx\times\cx$ $$\{x\in\cx : A_x\notin\ci\}\in\ci\Longrightarrow\{y\in\cx : A^y\notin\ci\}\in\ci$$

Natural examples of ideals fulfilling Fubini property are the ideal of null sets $\bbl$ (by Fubini theorem) and the ideal of meager sets $\bbk$ (by Kuratowski-Ulam theorem).

By definition we can obtain the following properties:

Assume that $\ci\perp \cj.$

1.  There exist a $(\ci,\cj)$ - Luzin set.

2.  If $L$ is a $(\ci,\cj)$ - Luzin set then $L$ is not $(\cj,\ci)$ - Luzin set.

(Part 1) By the definition of $\ci\perp\cj$ we can find two sets $I\in\ci$ and $J\in\cj$ such that $I\cup J=\cx.$ We will show that $J$ is $(\ci,\cj)$ - Luzin set. $J$ is not in $\ci.$ Let us fix any set $A\in\ci.$ We have that $A\cap J\subseteq J\in\cj.$

(Part 2) By the definition of $\ci\perp\cj$ we can find two sets $I\in\ci$ and $J\in\cj$ such that $I\cup J=\cx.$ Assume that $L$ is $(\ci,\cj)$ - Luzin set and $(\cj,\ci)$ - Luzin set. We have that $$L\cap J\subseteq J\in \cj \text{ and } L\cap I\subset I\in \ci$$ By the property of being $(\cj,\ci)$ - Luzin set $$L\cap J\in\ci.$$ So $L= (L\cap J)\cup (L\cap I)\in \ci.$ what is a contradiction with being $(\ci, \cj)$ - Luzin set.

We will try to find a wide class of forcings which preserves the property of being $(\ci, \cj)$ - Luzin set. We are mainly interested in so called definable forcings (see [@zapletal]). Let us recall that $\bbp$ is definable forcing if $\bbp$ is of the form $\Borel(\cx)\setminus\ci,$ where $\cx$ and $\ci$ have absolute definition for standard transitive models of ZF of the same hight.

Existence of Luzin sets 
========================

Let us start with a theorem which under suitable assumptions guarantees existence of uncountably many pairwise different $(\ci,\cj)$ - Luzin sets.

Assume that $\kappa=cov(\ci)=cof(\ci)\le non(\cj).$ Let $\cf $ be a family of functions from $\cx$ to $\cx.$ Assume that $|\cf|\le\kappa.$ Then we can find a sequence $(L_\alpha)_{\alpha<\kappa}$ such that

1.  $L_\alpha$ is $(\kappa,\ci,\cj)$ - Luzin set,

2.  for $\alpha\neq\beta,\ L_\alpha$ is not equivalent to $L_\beta$ with respect to the family $\cf.$

Let us enumerate the family $\cf$: $$\cf=\{ f_\alpha:\alpha<\kappa\}.$$ Now, let us enumerate Borel base of ideal $\ci$: $$\cb_\ci=\{ B_\alpha:\;\alpha<\kappa\}.$$

Now without loss of generality we can assume that $$(\forall f\in\cf)(\forall \lambda<\kappa) ( \kappa\le |f[(\bigcup\nolimits_{\xi<\lambda}B_\xi)^c]| )$$ Indeed, since $cov(\ci)=\kappa$ a set $(\bigcup\nolimits_{\xi<\lambda}B_\xi)^c$ is not in the ideal $\ci.$ If the function $f$ does not have the above property and $L$ is a $(\ci,\cj)$-Luzin set then $$f[L]=f[L\cap \bigcup\nolimits_{\xi<\lambda}B_\xi]\cup f[L\cap (\bigcup\nolimits_{\xi<\lambda}B_\xi)^c]$$ and both sets has cardinality less than $\kappa.$ So $f[L]$ is not $(\ci,\cj)$-Luzin set.

By induction we will construct the family $\{x^\eta_{\alpha,\zeta}: \eta,\zeta,\alpha<\kappa\}$ and $\{d^\eta_{\alpha,\zeta}: \eta,\zeta,\alpha<\kappa\}$ such that $$d^\eta_{\alpha,\zeta}=f_\zeta(x^\eta_{\alpha,\zeta})$$ and for any different $\eta,\eta'<\kappa$ $$\{x^\eta_{\alpha,\zeta}: \zeta,\alpha<\kappa\} \cap \{d^{\eta'}_{\alpha,\zeta}: \zeta,\alpha<\kappa\}=\emptyset$$ and $$x_{\alpha,\zeta}^\eta\in\cx\setminus \left(\{d^\eta_{\xi,\zeta}:\;\eta,\xi,\zeta<\alpha\}\cup \{x^\eta_{\xi,\zeta}:\;\eta,\xi,\zeta<\alpha\}\cup\bigcup_{\xi<\alpha }B_\xi\right)$$ for every $\eta,\zeta<\alpha.$

Assume that we are in $\alpha$-th step of construction. Fix $\eta,\zeta<\alpha$. It means that we have constructed the following set $$Old=\{ x^\lambda_{\beta,\xi},d^\lambda_{\beta,\xi}:\; \beta,\xi,\lambda<\alpha\}\cup \{ x^\lambda_{\alpha,\xi},d^\lambda_{\alpha,\xi}:\; \lambda<\eta\lor (\lambda=\eta\land \xi<\zeta)\}.$$ Since $|f_\zeta[(\bigcup_{\xi<\alpha} B_\xi)^c]|\ge\kappa$ and $|Old|<\kappa$ we get that $$|f_\zeta[(\bigcup_{\xi<\alpha} B_\xi\cup Old)^c]|\ge\kappa.$$ That’s why we can find $$d^\eta_{\alpha,\zeta}\in f_\zeta[(\bigcup_{\xi<\alpha} B_\xi\cup Old)^c]\setminus Old.$$ Let $x^\eta_{\alpha,\zeta}$ be such that $d^\eta_{\alpha,\zeta}=f_\zeta(x^\eta_{\alpha,\zeta})$. In this way we can finish the $\alpha$-th step of construction.

Now, let us define $L_\alpha=\{x^\alpha_{\xi,\zeta} : \xi,\zeta<\kappa\}.$

Let us check that $L_\alpha$ is $(\ci,\cj)$ - Luzin set. Indeed, if $A\in \ci$ then there exists $\beta<\kappa$ s.t. $A\subset B_\beta$. Then we have $$A\cap L_\alpha\subset B_\beta\cap L_\alpha=
B_\beta\cap \{x^\alpha_{\xi,\zeta}:\;\xi,\zeta<\beta\}
\subseteq \{x^\alpha_{\xi,\zeta}:\;\xi,\zeta<\beta\}\in \cj$$ because $|\{x^\alpha_{\xi,\zeta}:\;\xi,\zeta<\beta\}|\le|\beta|<\kappa\le non(\cj).$ What is more, for every function $f=f_\alpha\in\cf$ and every $\beta\neq\gamma$ we have that $$\kappa\le |f[L_\gamma]\setminus L_\beta|$$ because $\{ d^{\gamma}_{\xi,\alpha}:\alpha<\xi<\kappa\}\subseteq f[L_\gamma]\setminus L_\beta$. So $L_\beta\neq f[L_\gamma].$

In fact we haved proved a little stronger result.

\[stronger\] Assume that $\kappa=cov(\ci)=cof(\ci)\le non(\cj).$ Let $\cf $ be a family of functions from $\cx$ to $\cx.$ Assume that $|\cf|\le\kappa.$ Then we can find a sequence $(L_\alpha)_{\alpha<\kappa}$ such that

1.  $L_\alpha$ is $(\kappa,\ci,\cj)$ - Luzin set,

2.  for $\alpha\neq\beta$ and $f\in\cf$ we have that $\kappa\le |f[L_\alpha]\vartriangle L_\beta|.$

Let us notice that for every ideal $\ci$ we have the inequality $cov(\ci)\le cof(\ci).$ This gives the following corollary.

\[conti\] If $2^\omega=cov(\ci)=non(\cj)$ then there exists continuum many different $(\ci,\cj)$ - Luzin sets which aren’t Borel equivalent.

In particular, if CH holds then there exists continuum many different $(\omega_1,\ci,\cj)$ - Luzin sets which aren’t Borel equivalent.

We can extend above corollary to a wilder class of functions - namely, $\ci$-measurable functions.

\[conti2\] If $2^\omega=cov(\ci)=non(\cj)$ then there exists continuum many different $(\ci,\cj)$ - Luzin sets which aren’t equivalent with respect to all $\ci$-measurable functions.

In particular, if CH holds then there exists continuum many different $(\omega_1,\ci,\cj)$ - Luzin sets which aren’t equivalent with respect to all $\ci$-measurable functions.

First, let us notice that if a function $f$ is $\ci$-measurable then there exists a set $I\in\ci\cap\Borel(\cx)$ such that $f\upharpoonright (\cx\setminus I)$ is Borel. Indeed, it is enough to consider a countable base $\{U_n\}_{n\in\omega}$ of topology of $\cx.$ Then $f^{-1}[U_n]=B_n\vartriangle I_n,$ where $B_n$ is Borel and $I_n$ is from the ideal $\ci.$ Now, put $I=\bigcup_{n\in\omega}I_n.$

So we can consider a family of partial Borel functions which domain is Borel set with complement in the ideal $\ci.$ This family is naturally of size continuum. So we can use Corollary \[conti\] and Remark \[stronger\] to finish the proof.

Now, let us concentrate on ideal of null and meager sets.

1.  Assume that $cov(\bbl)=2^\omega.$ There exists continuum many different $(2^\omega,\bbl,\bbk)$ - Luzin sets which aren’t equivalent with respect to the family of Lebesgue - measurable functions.

2.  Assume that $cov(\bbk)=2^\omega.$ There exists continuum many different $(2^\omega,\bbk,\bbl)$ - Luzin sets which aren’t equivalent with respect to the family of Baire - measurable functions.

Let us notice that the equality $cov(\bbl)=2^\omega$ implies that $2^\omega=cov(\bbl)=cof(\bbl)=non(\bbk).$ Similarly, the equality $cov(\bbk)=2^\omega$ implies that $2^\omega=cov(\bbk)=cof(\bbk)=non(\bbl)$ (see [@cichondiagram]). Corollary \[conti2\] finishes the proof.

Luzin sets and forcing
======================

Now, let us focus on the class of forcings which preserves being $(\ci,\cj)$-Luzin set. Let us start with a technical observation.

\[jacek\] Assume that $\ci$ has Fubini property. Suppose that $\bbp_\ci=\Borel(\cx)\setminus \ci$ is a proper definable forcing. Let $B\in \ci$ be a set in $V^{\bbp_\ci}[G].$ Then $B\cap \cx^V\in \ci$.

Let $\dot{B}$ – name for $B$, $\dot{r}$ – canonical name for generic real, $C\subseteq \cx\times\cx$ - Borel set from the ideal $\ci.$ $C$ is coded in ground model $V$ and $B=C_{\dot{r}_G}.$

Now by Fubini property: $$\{ x:\;\; C^x\notin \ci\}\in \ci.$$ Let $x\in B\cap\cx^V$ then $V[G]\models x\in B$ $$0<\lbv x\in \dot{B}\rbv =\lbv x\in C_{\dot{r}}\rbv =\lbv (\dot{r},x)\in C\rbv=\lbv\dot{r}\in C^x\rbv =[C^x]_\ci$$ Then we have: $$B\cap\cx^V\subseteq\{ x: C^x\notin \ci\}\in \ci.$$ But the last set is coded in ground model because the set $C$ was coded in $V$.

Assume that $\omega<\kappa$ and $\ci, \cj$ are c.c.c. and have Fubini property. Suppose that $\bbp_\ci=\Borel(\cx)\setminus \ci$ and $\bbp_\cj=\Borel(\cx)\setminus \cj$ are definable forcings. Then $\bbp_\cj$ preserves $(\kappa,\ci,\cj)$ - Luzin set property.

Let $L$ be a $(\kappa,\ci,\cj)$ - Luzin set in $V$. In $V[G]$ take any $B\in \ci$ then $L\cap B\cap V=L\cap B$ but $L\cap B\in \ci$ in $V$ so $L\cap B\in \cj$ in $V$ by definition of $L.$ Finally, by Lemma \[jacek\] $$L\cap B=L\cap B\cap V\in \cj\text{ in } V[G].$$

Let $(\bbp,\le)$ be a forcing notion such that $$\{ B : B\in\ci\cap \Borel(\cx), B \text{ is coded in } V\}$$ is a base for $\ci$ in $V^\bbp[G].$ Assume that Borel codes for sets from ideals $\ci,\cj$ are absolute. Then $(\bbp,\le)$ preserve being $(\ci,\cj)$ - Luzin sets.

Let $L$ be a $(\ci,\cj)$ - Luzin set in ground model $V.$ We will show that $V^\bbp[G]\models L\text{ is }(\ci,\cj)\text{ - Luzin set}.$

Let us work in $V^\bbp[G].$ Fix $I\in\ci.$ $\ci$ has Borel base consisting of sets coded in $V$. So, there exists $b\in\omega^\omega\cap V$ such that $I\subseteq\#b\in\ci.$

By absoluteness of Borel codes from $\ci$ we have that $V\models\#b\in\ci.$ $L$ is a $(\ci,\cj)$ - Luzin set in the model $V.$ So, there is $c\in \omega^\omega\cap V$ which codes Borel set from the ideal $\cj$ such that $V\models L\cap\#b\subseteq\#c.$ By absoluteness of Borel codes from $\cj$ we get that $$V^\bbp[G]\models L\cap B\subseteq L\cap\#b\subseteq\#c\in\cj,$$ what proves that $L$ is a $(\ci,\cj)$ - Luzin set in generic extension.

The above theorem gives us a series of corollaries.

\[same-reals\] Let $(\bbp,\le)$ be any forcing notion which does not change the reals i. e. $(\omega^\omega)^V=(\omega^\omega)^{V^\bbp[G]}.$ Assume that Borel codes for sets from ideals $\ci,\cj$ are absolute. Then $(\bbp,\le)$ preserve being $(\ci,\cj)$ - Luzin sets.

\[wniosek1\] Assume that $(\bbp,\le)$ is a $\sigma$-closed forcing and Borel codes for sets from ideals $\ci,\cj$ are absolute. Then $(\bbp,\le)$ preserve $(\ci,\cj)$ - Luzin sets.

\[iteracja\] Let $\lambda\in On$ be an ordinal number. Let $\bbp_\lambda=\< (P_\alpha,\dot{Q}_\alpha):\;\;\alpha<\lambda\>$ be iterated forcing with countable support. Spouse that

1.  for any $\alpha<\lambda$ $P_\alpha\force \dot{Q}_\alpha - \sigma\text{ closed }$,

2.  Borel codes for sets from ideals $\ci, \cj$ are absolute,

then $\bbp_\lambda$ preserve $(\ci, \cj)$ - Luzin sets.

Our forcing $\bbp_\lambda$ is $\sigma$-closed because it is countable support iteration of $\sigma$ -closed forcings. So, we can apply Corollary \[wniosek1\] to finish the proof.

Now, let us consider some properties of countable support iteration connected with preservation of some relation. We will follow notation given by Goldstern (see [@goldi]).

First, let us consider measure case. Let $\Omega$ is a family of clopen sets of Cantor space $2^\omega$ and $$C^{random}=\{ f\in \Omega^\omega:(\forall n\in\omega) \mu(f(n))<2^{-n}\}$$ with discrite topology. If $f\in C^{random}$ then let us define the following set $A_f=\bigcap_{n\in\omega}\bigcup_{k\ge n} f(k)$.

Now, we are ready to define the following relation $\sqsubseteq=\bigcup_{n\in\omega} \sqsubseteq_n$ where $$(\forall f\in C^{random})(\forall g\in 2^\omega) (f\sqsubseteq_n g\iff (\forall k\ge n)\; g\notin f(k)).$$

Definition of the notion of preservation of relation $\sqsubseteq^{random}$ by forcing notion $(\bbp,\le)$ can be found in paper [@goldi]. Let us focus on the following consequence of that definition.

\[outer\] If $(\bbp,\le)$ preserves $\sqsubseteq^{random}$ then $\bbp\force \mu^*(2^\omega\cap V)=1$.

Now, we say that forcing notion $\bbp$ preserves outer measure iff $\bbp$ preserves $\sqsubseteq^{random}$.

It is well known that Laver forcing preserves some stronger property than $\sqsubseteq^{random}$ (see [@judah]). So, Laver forcing preserves outer measure.

In [@goldi] we can find the following theorem:

\[goldi\_iteration\] Let $\bbp_\lambda=((P_\alpha,Q_\alpha):\;\alpha<\gamma)$ be any countable support iteration such that $$(\forall \alpha<\gamma)\; P_\alpha\force Q_\alpha \text{ preserves }\sqsubseteq^{random}$$ then $\bbp_\gamma$ preserves the relation $\sqsubseteq^{random}$.

\[random\_luzin\] Assume that $\bbp$ is a forcing notion which preserves $\sqsubseteq^{random}$. Then $\bbp$ preserves being $(\bbl, \bbk)$-Luzin set.

Assume that $V\models L \mbox{ is } (\bbl,\bbk)\mbox{-Luzin set.}$ Let us work in $V^\bbp[G].$ Take any null set $A\in\bbl.$ Then there is a null set $B$ in ground model such that $A\cap V\subseteq B.$

Indeed, let us assume that there is no such $B\in V.$ Then without loss of generality $(2^\omega\setminus A)\cap V\in\bbl.$ But $A\in\bbl$ then we have that $2^\omega\cap V\subset A\cup ((2^\omega\setminus A)\cap V)$ which is a null set. But by Fact \[outer\] $\mu^*(2^\omega\cap V)=1.$ So we have a contradiction.

Then intersection $A\cap L\subseteq B\cap L\in\bbk$ is a meager set in ground model. Then by absolutnes of borel codes of meager sets the set $A\cap L$ is a meager set what finishes the proof.

In constructible universe $L$ let us consider the countable forcing iteration $P_{\omega_2}=((P_\alpha,Q_\alpha):\alpha<\omega_2)$ of the length $\omega_2$ as follows, for any $\alpha<\omega_2$

-   if $\alpha$ is even then $P_\alpha\force ''Q_\alpha \text{ is random forcing}''$,

-   in odd case $P_\alpha\force ``Q_\alpha\text{ is Laver forcing}''$.

Previously we noticed that both random and Laver forcing, preserves $\sqsubseteq^{random}$ and then by Theorem \[goldi\_iteration\] $P_{\omega_2}$ preserves relation $\sqsubseteq^{random}.$ By Theorem \[random\_luzin\] the $(\bbl,\bbk)$-Luzin sets are preserved by our iteration $P_{\omega_2}$. Moreover, in generic extension we have $cov(\bbl)=\omega_2$ and $2^\omega=\omega_2$ (for details see [@goldi]).

Asuume that in the ground model $A$ is $(\bbl,\bbk)$-Luzin set with outer measure equal to one. Then in generic extension it has outer measure one and $|A|=\omega_1.$ So, it does not contain any Lebesgue positive Borel set. Thus $A$ is completely $\bbl$-nonmeasurable set.

The analogous machinery can be used for ideal of meager sets $\bbk$. Let us recall the necessary definitions (see [@goldi]).

Let $C^{Cohen}$ be set of all functions from $\omega^{<\omega}$ into itself. Then $\sqsubseteq^{Cohen}=\bigcup_{n\in\omega} \sqsubseteq_n^{Cohen}$ and for any $n\in\omega$ let $$(\forall f\in C^{Cohen})(\forall g\in \omega^\omega)( f\sqsubseteq_n^{Cohen} g\text{ iff }
(\forall k<n)( g\upharpoonright k^{\frown} f(g\upharpoonright k)\subseteq g)).$$

Then finally we have the following theorem:

\[cohen\_luzin\] Assume that $\bbp$ is a forcing notion which preserves $\sqsubseteq^{Cohen}$. Then $\bbp$ preserves being $(\bbk, \bbl)$-Luzin set.

The another preservation theorem which is due to Shelah (see [@shelah] and also [@schliri]) is as follows

\[shelah\_iteration\] Let $\bbp_\lambda=((P_\alpha,\dot{Q}_\alpha):\;\alpha<\lambda)$ be any countable support iteration such that $(\forall \alpha<\gamma)$ $P_\alpha\force Q_\alpha \text{is proper } $ and $$P_\alpha\force Q_\alpha\force \text{every new open dense set contains old open dense set}$$ then $\bbp_\lambda \force \text{ every new open dense contains old open dense set}$.

We can easily derive

Let $\bbp_\lambda=((P_\alpha,\dot{Q}_\alpha):\;\alpha<\lambda)$ be any countable support iteration such that $(\forall \alpha<\lambda)$ $P_\alpha\force Q_\alpha \text{is proper } $ and $$P_\alpha\force Q_\alpha\force \text{every new open dense set contains old open dense set}$$ Then $\bbp_\lambda$ preserves being $(\bbk,\bbl)$-Luzin set.

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