If $\{f_n\}$ is sequence of measurable functions on $X$, then $\{x: \lim f_n(x) \text{ exists}\}$ is a measurable set.
69857
Is the set $A=\{x\in E|\lim_{n \to\infty} f_n(x)\ exists\ (finitely)\}$ measurable?
40297
Proof of uniform limit of Continuous Functions
153854
$X_{n}$ converges in probability, $f$ is continuous, $f(X_{n})$ converges in probability
129087
Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$
140445
Sequence of continuous functions whose pointwise limit is discontinuous
80
Convergence in measure implies convergence almost everywhere of a subsequence
106960
Proof to sequences in real analysis
147546
$f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable
66391
Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable
22597
Composition of measurable function and continuous function
81843
Sequences of sets property
151707
Fatou's lemma and measurable sets
157821
Let $S$ be a finite set. Show that any function $f: S \rightarrow \mathbb{R}^n$ is continuous on S. What if $S$ is countably infinite?
48891
Show that: lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$
35514
If $\{f_n\}$ converges uniformly and each $f_n$ is bounded, show that $\exists M>0$ s.t. $|f_n(x)| \leq M$
53663
outer measure and measurable sets
135656
Why non-measurable sets exist?