# How to prove limit of measurable functions is measurable

I need help to prove the following theorem

Suppose $f$ is the pointwise limit of a sequence of $f_n$, $n = 1, 2, \cdots$, where $f_n$ is a Borel measurable function on $X$. Then $f$ is Borel measurable on $X$.

My idea is to use the standard definition like for every $c$,$\{x:f(x)<c\}$ is Borel measurable. But got stuck as how to do it for sequence of $f_n$.

• Check Theorem 1.14 of Rudin's RCA, and also its Corollary (a). Commented Mar 1, 2021 at 4:38

First, we prove that $$\limsup\limits_{n\to\infty\space k\geqslant n}f_k$$ and $$\liminf\limits_{n\to\infty\space k\geqslant n}f_k$$ are measurable.

By definition $$\limsup\limits_{n\to\infty\space k\geqslant n}f_k=\inf\limits_{n\geqslant1}\sup\limits_{\space k\geqslant n}f_k$$ $$\liminf\limits_{n\to\infty\space k\geqslant n}f_k=\sup\limits_{n\geqslant1}\inf\limits_{\space k\geqslant n}f_k$$

Since \begin{align} \inf_{n\geqslant 1} \sup_{k\geqslant n} f_k(x) \leqslant c &\iff \forall \epsilon>0,\: \exists n\geqslant 1: \quad \sup_{k \geqslant n} f_k(x)< \inf \sup_{k\geqslant n} f_k(x)+\epsilon \leqslant c+\epsilon \\ &\iff \forall j\geqslant 1,\: \exists n\geqslant 1: \quad \sup_{k \geqslant n} f_k(x) \leqslant c+\frac1{j} \\ &\iff \forall j\geqslant 1,\: \exists n\geqslant 1, \:\forall k\geqslant n: \quad f_k(x) \leqslant c+\frac1{j} \\ &\iff x\in\bigcap\limits_{j\geqslant 1}\bigcup\limits_{n\geqslant 1}\bigcap\limits_{k\geqslant n}\left\{x:f_k(x) \leqslant c+\frac1{j}\right\} \end{align} We have $$\{x:\inf\limits_{n\geqslant1}\sup\limits_{\space k\geqslant n}f_k\leqslant c\}=\bigcap\limits_{j\geqslant 1}\bigcup\limits_{n\geqslant 1}\bigcap\limits_{k\geqslant n}\left\{x:f_k(x) \leqslant c+\frac1{j}\right\}\tag1$$ Similarly \begin{align} \sup_{n\geqslant 1} \inf_{k\geqslant n} f_k(x) \leqslant c &\iff \forall n\geqslant 1: \quad \inf_{k \geqslant n} f_k(x)\leqslant c \\ &\iff \forall \epsilon>0,\:\forall n\geqslant 1, \:\exists k\geqslant n: \quad f_k(x)<\inf_{k \geqslant n} f_k(x)+\epsilon\leqslant c+\epsilon \\ &\iff \forall j\geqslant 1,\: \forall n\geqslant 1, \:\exists k\geqslant n: \quad f_k(x) \leqslant c+\frac1{j} \\ &\iff x\in\bigcap\limits_{j\geqslant 1}\bigcap\limits_{n\geqslant 1}\bigcup\limits_{k\geqslant n}\left\{x:f_k(x) \leqslant c+\frac1{j}\right\} \end{align} So $$\{x:\sup\limits_{n\geqslant1}\inf\limits_{\space k\geqslant n}f_k\leqslant c\}=\bigcap\limits_{j\geqslant 1}\bigcap\limits_{n\geqslant 1}\bigcup\limits_{k\geqslant n}\left\{x:f_k(x) \leqslant c+\frac1{j}\right\}\tag2$$ Since both $$(1)$$ and $$(2)$$ are measurable $$\limsup\limits_{n\to\infty\space k\geqslant n}f_k \quad\text{ and }\quad\liminf\limits_{n\to\infty\space k\geqslant n}f_k\quad\text{}$$ are measurable. Since $$\lim\limits_{n\to\infty\space}f_n=f$$, $$f=\limsup\limits_{n\to\infty\space k\geqslant n}f_k=\liminf\limits_{n\to\infty\space k\geqslant n}f_k$$ So $$f$$ is measurable.

• Good answer. Now: what if the range is not the reals? I think the OP intended the reals, but he didn't say. Commented Jun 19, 2015 at 20:11
• Corollary: If each $f_n :R\to R$ is Lebesgue measurable and $(f_n)_{n \in N}$ converges point-wise almost everywhere to $f$ then $f$ is measurable. Commented Nov 7, 2015 at 7:34
• Why wouldn't it have been enough to show that either the lim sup or lim inf is measurable and that $f$ is equal to one of them? Commented Sep 28, 2022 at 22:29
• I want to give more detail of the proof. Commented Sep 29, 2022 at 0:18

As a more "direct" proof, you could also note that $$\{x\in X\mid f(x)>a\}=\bigcup_{m=1}^\infty\bigcup_{N=1}^\infty \bigcap_{n=N}^\infty \left\lbrace x\in X :f_n(x)>a+\frac{1}{m}\right\rbrace$$

• Can you explain right hand side? Commented Sep 21, 2022 at 13:37
• @Jonathen I think that $x\in X$ is an element of the RHS if and only if there exists an $m$ such that $$a+\frac 1m<f_n(x)$$for almost all $n$. Commented Dec 18, 2022 at 10:07

Hint

When in trouble, go big: prove that $$\limsup_n f_n=\inf_n\sup_{m\ge n}f_m$$ is Borel-measurable.

So you only need to prove that $\sup_n f_n$ (and $\inf_nf_n$) is (are) Borel-measurable whenever the $f_n$-s are Borel-measurable.