If $f_n$ and $f$ are integrable, $f_n \to f$ a.e., and $\int |f_n| \to \int |f|$, then $\int |f_n-f| \to 0$.
9613
If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$
95626
$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.
88690
$\int_a^b f(x)\mathrm dx=0$ for all $-\infty<a<b<\infty$ a,b rational implies $f=0$ a.e
119200
When does $\lim_{n\to\infty}f(x+\frac{1}{n})=f(x)$ a.e. fail
50472
If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$
130161
Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
32525
If $f(1)=1$, then is it true that $f(n)=n$ for all $n \in \mathbb{N}\cup\{0\}$.
148640
How to prove that if $\forall x \in (a,b)$ Lebesgue integral $\int_{(a,x)}fd\lambda=0$, then $f(x)=0$ $\lambda$-almost everywhere?
144322
Prove for bounded $f$, if $f:[a,r] \to \mathbb{R}$ is Riemann-Integrable for $r \in [a,b)$ then $f:[a.b] \to \mathbb{R}$ is Riemann-Integrable
72847
Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
94564
If $f$ is bounded and $\lim_{x \to c} g(x) = 0$, then $\lim_{x \to c}g(x)f(x) = 0$
66504
Proof that $f$ is Lebesgue-integrable and calculate its integral
94066
Prove that: {$f_n$} converges unifromly to $f$ on $E$ as $n \rightarrow \infty$ if and only if $\lim_{n\rightarrow \infty} ||f_n-f||_u = 0$
77106
If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ ...
31319
An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.
25996
Using the definition of Riemann integrability, prove that $f $is Riemann integrable on $[0,1]$
133442
An integrable Functions is almost everywhere finite