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Consider a locally bounded function $f: X \times W \rightarrow X$, where $X \subseteq \mathbb{R}^n$, $W \subseteq \mathbb{R}^m$, such that

for all $x \in X$ the function $w \mapsto f(x,w)$ is (Borel) measurable;

Consider a locally bounded, (Borel) measurable, function $g: W \rightarrow X$.

Say if the function

$$ (w,v) \mapsto f( g(w), v ) $$

is (Borel) measurable as well.

Notes: this question differs from both this and that post.

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    [Here](http://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch3.pdf): if $f: \mathbb{R} \rightarrow \mathbb{R}$ is Borel measurable and $g: \mathbb{R}^m \rightarrow \mathbb{R}$ is Borel measurable then $f \circ g$ is Borel measurable as well. Can we generalize that?2012-06-29
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    Also in $\mathbb{R}^n$, the composition of two Borel mappings is a Borel mapping as well [Bogachev - Measure Theory, pag. 146].2012-06-29

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What you are trying to prove does not hold. Let $V$ be the Vitali Set which is not measurable in $B(\mathcal{R})$. Let $f: R^{2} \mapsto R$, such that:

$$ f(x,y) = \left\{ \begin{array}{ll} 1 & \mbox{$x \in V$};\\ 0 & \mbox{$x \notin V$}.\end{array} \right. $$

Note that $f$ is locally bounded and for any $x \in R$, $y \mapsto f(x,y)$ is Borel measurable, since it is constant. Let $g$ be the identity function. Observe that $f(g(x),y) = f(x,y)$ which is not measurable in $(R^{2},B(\mathcal{R}^{2}))$.

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    Why $f$ is measurable?2012-06-29
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    Also, if $g$ is from $W$ to $X$ how can it be the identity?2012-06-29
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    f is not measurable... but you didn't say it had to be. Consider $W = X$, for this example.2012-06-29
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    I mean why for all $x \in X$ we have $w \mapsto f(x,w)$ measurable?2012-06-29
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    If $x \in A$, $w \mapsto f(x,w) = 1$ which is measurable. If $x \notin A$, $w \mapsto f(x,w) = 0$, which is measurable.2012-06-29
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    So in your example, $n=1$. Why is this example not contradicting [this](http://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch3.pdf) (after Definition 3.3)?2012-06-29
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    I see no contradiction. Can you write where you think the contradiction is?2012-06-29
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    For instance, what do you mean by $A \in X$ is not measurable? Any measurable/non-measurable set should be in $W$, not in $X$. No measures are defined for $X$.2012-06-29
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    I will edit the answer to make it more explicit.2012-06-29