Set $A\subset \mathbb{R}$ so that $\lambda(A\cap [a,b])>0$ and $\lambda([a,b]\setminus A)>0$.
38581
Construct a Borel set on R such that it intersect every open interval with non-zero non-"full" measure
95947
Cantor set + Cantor set =$[0,2]$
155311
How to get a set $E$ such that $0<\mu(E\cap I)<\mu(E)$ for all interval $I$?
65595
Let $\lambda(A)$ be the Lebesgue measure of $A$. There exists an interval $I$ such that $\lambda(E \cap I) / \lambda(I) < 1-\epsilon$
48162
Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?
90826
Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly
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 $\mathbb R^1$
69499
Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0\alpha$.
63538
Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$
76764
Find two closed subsets or real numbers such that $d(A,B)=0$ but $A\cap B=\varnothing$
20335
If $A\subset B$ then $\bar{A}\subset \bar{B}$
19289
complement of a set and its measure
46056
If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A
73927
Is $\mathbb{R}$ a subset of $\mathbb{R}^2$?
115606
Proofs involving $ (A\setminus B) \cup (A \cap B)$
44430
Is there an open dense set $S \subset [0,1]$ such that $m(S)<1$?
104411
sum of two connected subset of $\mathbb{R}$