Show that a borelien outer measure on a metric space is a metric outer mesure.
62034
If every open set of a metric space $(X, \rho)$ is $\mu^*$-measurable, then $\mu^*$ is a metric outer measure.
19773
Is finite additivity a property of the Lebesegue outer measure?
13848
Showing that if a subset of a complete metric space is closed, it is also complete
53425
Prove that there exists a Borel measurable function $h: \mathbb R \to \mathbb R$ such that $g= h\circ f$.
123039
If $0<\alpha < 1$ and $d(x,y)$ is metric, show that $((d(x,y))^{\alpha}$ is metric.
115219
If $E$ is Lebesgue measurable, show that there exists a closed set $F$ with $F \subset E$ and $m(E\setminus F)<\epsilon$
113227
Measure spaces are proper subsets
137257
The Supremum and Infimum of a sequence of measurable functions is measurable
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 $\mathbb R^1$
128598
How to prove this is a metric?
65840
Norm induced by metric
133152
(X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀𝑥 ∈ X, B(x,ε) is contained in some member of the cover.
97841
General property regarding outer measure for a nested sequence of sets (measurable or not).
144817
Proving Caratheodory measurability if and only if the measure of a set summed with the measure of its complement is the measure of the whole space.
118353
For set of positive measure $E$, $\alpha \in (0, 1)$, there is interval $I$ such that $m(E \cap I) > \alpha \, m(I)$
65809
Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric
114047
Metric $d(x,y)=\frac{|x-y|}{\sqrt{1+x^2}\sqrt{1+y^2}}$ on $\mathbb{R}$