Measurable Set $E$ so that $m(E\cap(a, b)) \gt 0$ for all $a, b \in \mathbb{R} $ and $m(E) \lt \infty$
3023
Construction of a Borel set with positive but not full measure in each interval
63413
For a set of positive measure there is an interval in which its density is high, $\mu(E\cap I)> \rho \mu(I)$
137203
Measurable cardinal existence condition
87216
Lebesgue non-measurable function
69273
Is there a compact subset of the irrationals with positive Lebesgue measure?
113778
Show that $E=\{(x,\alpha)\mid 0\leq \alpha<|f(x)|\}$ is measurable if $f$ is measurable.
41683
Pre-image of a measurable set A is always measurable?
47668
Show that $lim_{n \rightarrow \infty}m(O_n)=m(E)$ when $E$ is compact.
51260
Sum of sets of measure zero
156176
Lebesgue measurability of nearly identical sets
121367
Let $\alpha \in (0,1)$. Find a Borel subset $E$ of $[-1,1]$ s.t. $\lim_{r\to 0^{+}} \frac{m(E\cap [-r,r])}{2r}=\alpha.$
115219
If $E$ is Lebesgue measurable, show that there exists a closed set $F$ with $F \subset E$ and $m(E\setminus F)<\epsilon$
69499
Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0\alpha$.
152310
If $E$ has Lebesgue measure $0$, must there exist a translate such that $E\cap E+x=\varnothing$?
131793
Zero integral of measurable $f$ on every interval implies $f=0$?
134310
Minkowski sum of a positive Lebesgue measure set and $\mathbb{Q}$.
146707
Formally show that the set of continuous functions is not measurable