# Tag Info

Accepted

### 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.

Step 0: The inequality $\mu^{\ast}(A) \leqslant \mu^{\ast}(A\cap E) + \mu^{\ast}(A \cap E^C)$ for all $A \subset X$ follows directly from the subadditivity of outer measures. Step 1: For a ...
• 208k
Accepted

### How does the frequentist definition of probability work with non-measurable sets?

Initial answer: I don't have a rigorous answer for you, but let me point out that every subset of $[0,1]$ (measurable or not) has a rigorously defined inner and outer Lebesgue measure, both of which ...
• 30.3k
Accepted

### Let $E_1 \subset E_2$ both be compact and $m(E_1) = a, m(E_2) = b$. Prove there exists a compact set $E$ st $m(E) = c$ where $a < c < b$.

Let $E_1, E_2$ be two compact sets in $\mathbb R^d$. Let $f : \mathbb R \to \mathbb R$ be defined by $$f(t) = m(E_1 \cup (E_2 \cap \{ x_1+ \cdots + x_d \le t\})),$$ where $x_1, \cdots, x_d$ are the ...
• 16.2k
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### Proving the *Caratheodory Criterion* for *Lebesgue Measurability*

Note: I just realized that your statement of Carathéodory's criterion doesn't agree with the usual one, where one tests against any possible test $A$. If we show that it is possible to test using open ...
• 1,644
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• 121k
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### Show that $\mathcal A_1$ $\cap$ $\mathcal A_2$ is also a $\sigma$-algebra

As mentioned by @Reveillark, you shall need the definition of intersection of sets. Based on it, we can proceed as follows. Let $\Omega$ be a nonempty set and $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ ...
• 21.6k
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• 3,908
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### Continuous map from subset of $\mathcal{C}$ (Cantor) to non-measurable set.

You are absolutely correct, for the reason you gave, and the "proof" sketched is false. What we know is that $f(C)$ is a compact set of positive measure. And any set of positive measure contains a ...
• 106k