Let $f:\mathbb{R}\rightarrow[0,\infty]$ be measurable that $\int f(x)dx=0$, then $f(x)=0$ for a.e.
118468
$\int f=0$ implies $f=0$ almost everywhere
90540
What is wrong in this proof: That $\mathbb{R}$ has measure zero
129551
Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?
35713
Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<|\mathbb{R}|$, is it true that $X$ is null?
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
58361
Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$?
67292
Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$
33989
If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.
95260
Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.
92089
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
41434
Show that: $\lim \limits_{n \rightarrow+\infty} \int_{0}^{1}{f(x^n)dx}=f(0)$
71953
If $\int_0^1 f x^n dx=0, \forall n \in \mathbb{N} $, then $f\equiv 0.$
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
126227
continuity of a probability measure if $\mu (\{x\})=0$
33081
If $\int_{\Bbb R^2}|f(x+iy)|\,dx\,dy<\infty$ then prove that $f\equiv 0$