If $f$ is Lebesgue integrable, does $\int_{\mathbb R}|f|=\lim_{t\to \infty }=\int_{-t}^t |f|$?
125910
Continuity of Integration for a measurable function
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
68811
If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.
127921
Does there exist a function such that $\lim_{x \to a} f(x) = L$ for all $a \in \mathbb R$ but $f(x)$ is never $L$?
150896
Show that $\lim_{p \to \infty} \| f \|_p = \| f \|_\infty$
127868
If $f$ is nonnegative and integrable then $F(x) = \int_{-\infty}^x f$ is continuous.
20404
Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$
49929
If $ \lim_{n \to \infty} (2 x_{n + 1} - x_{n}) = x $, then is it true that $ \lim_{n \to \infty} x_{n} = x $?
13732
$||f||_p\to ||f||_\infty$ under general assumptions
130161
Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
66504
Proof that $f$ is Lebesgue-integrable and calculate its integral
157715
Proving $f$ is Lebesgue integrable iff $|f|$ is Lebesgue integrable.
124916
$f$ such that $\int_1^{\infty}f(x)dx$ converges, but not absolutely?
64797
Does an absolutely integrable function tend to $0$ as its argument tends to infinity?
20675
$L=\lim_{x\to\infty}(f(x)+f'(x))$ exists . Which of the following statements is\are correct?
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?
38831
If $\lim_{x\to+\infty}[f(x+1)-f(x)]= \ell,$ then $\lim\limits_{x\to+\infty}\frac{f(x)}x=\ell$.