I found this two version of Lusin theorem, without reference
Can someone know in any book i can found it :
Theorem 1: Let $(X,\beta,\mu)$ a measurable space satisfying:
$\bullet$ $X$ is a metric space locally compact,
$\bullet$ $\beta$ is a borelian $\sigma-$algebra.
$\bullet$ $\mu$ is a regular measure.
Suppose that $A\in \beta$ such that $\mu(A)<+\infty.$ Then, given $\varepsilon>0$ there exists $g\in C_0(X)$ such that
$\bullet$ $\mu(\{x\in X; g(x)\neq \chi_{A}(x)\})<\varepsilon;$
$\bullet$ $g(x)\in [0,1], x\in X.$
and the second:
Theorem 2:
$(i)$ $Y$ be a Hausdorff topological space with a countable bases;
$(ii)$ $(\Omega,\beta,\mu)$ a finite measurable space, where $\Omega\subset Y$;
$(iii)$ $f:\Omega\to \mathbb{R}$ a measurable function. Given $\varepsilon>0$, there exists a compact set $K\subset \Omega$ such that
$(I)$ $f$ is continuous on $K$;
$(II)$ $\mu(\Omega\setminus K)<\varepsilon.$