$M \subseteq \mathbb R$ exist, so that $\lambda^{1}(M \cap I)= \frac{1}{2}\lambda^{1}(I)$ for any bounded Interval $I\subseteq \mathbb R$.
37361
Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$?
22149
How to prove $dim(Ker (A- \lambda I))=dim (Ker(A^t- \lambda I))$?
117478
Prove that $\mathbb{R}$ and the interval $(0, \infty)$ have the same cardinality.
128453
If $A$ has two eigenvalues $\lambda _1, \lambda_2$ and $\dim (E_{\lambda_1})=n-1$, then $A$ is diagonalizable
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 $\mathbb R^1$
96931
Supremum and infimum of $\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$
137707
How to prove that $(A\cup B) \cap C \subseteq A\cup (B\cap C)$
81927
Example of closed and bounded in $\mathbb Q$ set that is not compact
157924
$\forall x\in\mathbb{N}[(\exists y\in\mathbb{N}(2\leq y\land y<x\land y\ |\ x))\to(\exists z\in\mathbb{N}(2\leq z\land z\leq\sqrt x\land z\ |\ x))]$
68490
For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$
126698
measurability of a function -equivalent conditions
30694
Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$
80827
Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...
73690
If $\mathbb R^3\setminus V$ connected where $V$ is the subspace generated by $\{(1,1,1),(0,1,1)\} $
94363
Prove: Any open interval has the same cardinality of $\Bbb R$ (without using trigonometric functions)
104234
What is the Hausdorff dimension of the set of irrationals on $\mathbb R^1$
61640
Prove c exists for f:[0,1] -> R