An example that shows the space of functions with N Luzin property is not closed under addition I'm looking for two functions $f,g $ with N Luzin property that $f+g $ fails N Luzin property. 
Is there any hint how to construct them?
 A: Here is an example of two Borel functions $f,g:\mathbb R\to \mathbb R$ having property (N) such that $f=g$ does not have property (N). It should certainly be possible to find continuous functions $f,g$ with these properties; but I dont immediately see how.
Let us denote by $C\subseteq [0,1]$ the usual triadic Cantor set. It is well known that $C$ has measure $0$  and yet $C+C$ contains the interval $[0,1]$. We need the following
Fact. There exist two Borel functions $\alpha, \beta : [0,1]\to \mathbb R$ such that $\alpha$ and $\beta$ take their values in $C$ and $\alpha(x)+\beta(x)=x$ for every $x\in [0,1]$.
To prove the Fact, we need to show that there exists a Borel function $\alpha:[0,1]\to\mathbb R$
 taking its values in $C$ such that $x-\alpha(x)\in C$ for all $x\in [0,1]$. To show this, consider for any $x\in [0,1]$ the set $E_x:=\{ \alpha\in C;\; x-\alpha\in C\}$. This is a closed subset of $C$, hence a compact set, and $E_x\neq\emptyset$ because $C+C$ contains $[0,1]$. So we may define $\alpha(x):=\min \, E_x$. The function $\alpha:[0,1]\to \mathbb R$ is Borel (in fact, lower semi-continuous), because for any $c\in\mathbb R$ we have
$$\alpha^{-1}\bigl( (-\infty ,c]\bigr)=\{ x\in [0,1];\; \exists \alpha\in C\;:\; \alpha\leq c\;{\rm and}\; x-\alpha\in C\}\, ,$$
which is easily seen to be a closed set (by compactness of $C$). So the function $\alpha$ works by its very definition.
Consider now any Borel function $\phi:\mathbb R\to [0,1]$ which does not have property (N). Let $\alpha,\beta$ be the functions given by the Fact, and define
$$f(t):=\alpha(\phi(t))\qquad{\rm and}\qquad g(t):=\beta(\phi(t))\, .$$
Then $f,g$ are Borel, and they obviously have property (N) because their range is contained in $C$, which has measure $0$. But $f+g$ is equal to $\phi$, so it does not have property (N).
