Does a function being Lebesgue measurable imply it is integrable? ($f\colon [a,b]\to\Bbb C$)
63309
How to find a measurable but not integrable function or a positive integrable function?
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.
52632
$f\colon(0,\infty)\to \mathbb R$ be continuous ; $f(x)\le f(nx) , \forall n \in \mathbb N , \forall x >0$ , then $\lim_{x\to \infty} f(x)$ exists?
115878
Continuous unbounded but integrable functions
126698
measurability of a function -equivalent conditions
88418
Proof of a theorem saying that we can make a measurable function continuous by altering it by a set of arbitarily small measure
88610
The set of convergence of a sequence of measurable functions is measurable
157715
Proving $f$ is Lebesgue integrable iff $|f|$ is Lebesgue integrable.
112323
Showing that $f = 0 $ a.e. if for any measurable set $E$, $\int_E f = 0$
69857
Is the set $A=\{x\in E|\lim_{n \to\infty} f_n(x)\ exists\ (finitely)\}$ measurable?
143699
Bounded measurable function and integral with charcteristic function
146707
Formally show that the set of continuous functions is not measurable
139612
Follow-up regarding right-continuous $f:\mathbb{R} \to\mathbb{R}$ is Borel measurable
13765
Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?
125190
Convergence to function that is not measurable
78572
Why do we restrict the definition of Lebesgue Integrability?
92638
Approximation of measurable function by simple functions