Is composition of measurable functions measurable?
We know that if $ f: E \to \mathbb{R} $ is a Lebesgue-measurable function and $ g: \mathbb{R} \to \mathbb{R} $ is a continuous function, then $ g \circ f $ is Lebesgue-measurable. Can one replace the continuous function $ g $ by a Lebesgue-measurable function without affecting the validity of the previous result?
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$\begingroup$Here is the standard example:
Let $f\colon [0,1]\to [0,1]$ be the Cantor–Lebesgue function. This is a monotonic and continuous function, and the image $f(C)$ of the Cantor set $C$ is all of $[0,1]$. Define $g(x) = x + f(x)$. Then $g\colon [0,1] \to [0,2]$ is a strictly monotonic and continuous map, so its inverse $h = g^{-1}$ is continuous, too.
Observe that $g(C)$ measure one in $[0,2]$: this is because $f$ is constant on every interval in the complement of $C$, so $g$ maps such an interval to an interval of the same length. It follows that there is a non-Lebesgue measurable subset $A$ of $g(C)$ (Vitali's theorem: a subset of $\mathbb{R}$ is a Lebesgue null set if and only if all its subsets are Lebesgue measurable).
Put $B = g^{-1}(A) \subset C$. Then $B$ is a Lebesgue measurable set as a subset of the Lebesgue null set $C$, so the characteristic function $1_B$ of $B$ is Lebesgue measurable.
The function $k = 1_B \circ h$ is the composition of the Lebesgue measurable function $1_B$ and and the continuous function $h$, but $k$ is not Lebesgue measurable, since $k^{-1}(1) = (1_B \circ h)^{-1}(1) = h^{-1}(B) = g(B) = A$.
$\endgroup$ 2 $\begingroup$This Wikipedia article may be what you are looking for.
According to the article, a function $ f: \mathbb{R} \to \mathbb{R} $ is said to be Lebesgue-measurable if and only if for every Borel-measurable subset $ B $ of $ \mathbb{R} $, its pre-image $ {f^{\leftarrow}}[B] $ is a Lebesgue-measurable subset of $ \mathbb{R} $. Therefore, because we are dealing with two different $ \sigma $-algebras of $ \mathbb{R} $ here, the composition of two Lebesgue-measurable functions is not necessarily Lebesgue-measurable. This is precisely what GEdgar and AD. have mentioned.
$\endgroup$ $\begingroup$This details the Ilya comment:
Yes. In general let $(E_1,\mathcal T_l) ,(E_2,\mathcal T_2),(E_3,\mathcal T_3)$ be three measurable spaces and $f:E_1 \to E_2$ and $g:E_2 \to E_3$ measurables functions.If $X \in \mathcal T_3$, then , by measurability of $g$ we have : $g^{-1}(X) \in \mathcal T_2$ and by measurability of $f$ we have $f^{-1}(g^{-1}(X)) \in \mathcal T_1$. Since $(g \circ f)^{-1} (X)=f^{-1} ( g^{-1}(X))$ we have :$$\forall X\in \mathcal T_3 \quad (g \circ f)^{-1} (X) \in \mathcal T_1$$
This gives measurability of $g \circ f$
As an addition to the other answers, I think it is worth noting that in terms of imitating the relation between a topology and a continuous function when thinking of the relation between a $\sigma$-algebra and a measurable function, we can only go so far as having $g$ Borel measurable.
Let $\mathcal{T}$ be the open sets, $\mathcal{B}$ be the Borel sets and $\mathcal{M}$ be the Lebesgue measurable sets of the real line, respectively. Then we have that $\mathcal{T}\subset\mathcal{B}\subset\mathcal{M}$.
We call $f:E\to\mathbb{R}$ continuous if $f^{-1}(\mathcal{T})\subseteq\mathcal{T}$, Borel measurable if $f^{-1}(\mathcal{T})\subseteq\mathcal{B}$ and Lebesgue measurable if $f^{-1}(\mathcal{T})\subseteq\mathcal{M}$, respectively. Accompanying the inclusions of the classes of sets, we have that every continuous function is Borel measurable and every Borel measurable function is Lebesgue measurable.
It turns out that $f$ is Borel iff $f^{-1}(\mathcal{B})\subseteq\mathcal{B}$, which is analogous to the continuous case, but we don't have $f^{-1}(\mathcal{M})\subseteq\mathcal{M}$ in general for Lebesgue measurable functions $f$. The best we can do is that $f$ is Lebesgue measurable iff $f^{-1}(\mathcal{B})\subseteq\mathcal{M}$.
Consequently if $f$ is Lebesgue measurable and $g$ is Borel measurable, then $g\circ f$ is Lebesgue measurable (just like in the continuous case), and if $f,g$ are both Borel measurable then $g\circ f$ is Borel measurable (again, just like in the continuous case), but if both $f,g$ are Lebesgue measurable we can cook up examples where $g\circ g$ is not Lebesgue measurable. (For the sake of completeness, if $g$ is Lebesgue and $f$ is Borel then $g\circ f$ need not be Lebesgue: in the standard example above f is a homeomorphism).
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