Constructing a Borel set A on R such that $0<m(A \cap I) < m(I)$ for all intervals $I$.
3023
Construction of a Borel set with positive but not full measure in each interval
92571
Sets with no asymptotical density over $\mathbb N$
35631
$H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
69499
Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0\alpha$.
59424
Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
36631
Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?
115606
Proofs involving $ (A\setminus B) \cup (A \cap B)$
64320
Prove that if $0<a<Pr(A)$ then there exists $B\subset A$ such that $Pr(B)=a$
143586
Is $[0,1]$ a union of family of disjoint closed intervals?
95294
Dense subset of $C[0,1]$ which preserves rationals
3896
A closed set in $(A, \mathcal T_A)$ that is not closed in $(X, \mathcal T)$
19452
Restriction of a Lebesgue integral to a subset of a measurable set.
43535
show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$
93684
Show that $\mu (E \setminus A)>0$ if $A$ is a nonmeasurable subset of $X$ and $A \subseteq E$, $E$ is measurable.
121367
Let $\alpha \in (0,1)$. Find a Borel subset $E$ of $[-1,1]$ s.t. $\lim_{r\to 0^{+}} \frac{m(E\cap [-r,r])}{2r}=\alpha.$
69273
Is there a compact subset of the irrationals with positive Lebesgue measure?
154621
A set with a finite integral of measure zero?