# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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67 views

### Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
1answer
42 views

### Show that is a probability space

Let $\Omega:= \{(x,y) \in \mathbb{R^2}:0<x,y \leq 1 \}$, let $\mathcal{F}$ be the collection of sets of $\Omega$ such that $$\mathcal{F}:= \{(x,y) \in \mathbb{R^2}:x \in A,0<y \leq 1 \}$$ ...
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55 views

### If equality of dual space of a Banach spaces implys the equality of pre-duals?

Assume $X_1$ and $X_2$ are two Banach Spaces such that $X_1\subset X_2$, i.e., the element belongs to $X_1$ belongs to $X_2$. No assumption on norms. Then I would expect that the dual space of them ...
1answer
51 views

### Cumulative distribution function implication

How can I prove the following: Let $X$ and $Y$ be two random variables. Suppose that their cumulative distribution functions satisfies $F_X(x)=F_Y(x)$ for all $x$. How can I show that $X$ and $Y$ are ...
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71 views

### Extension of an additive function

Let $X$ be a finite set, $S\subset \mathcal P(X)$ such that: $1) X\in S$, $2) A,B\in S, A\cap B=\emptyset \Rightarrow A\sqcup B\in S$ and $3) A,B\in S, A\subset B \Rightarrow B\setminus A \in S$ ...
1answer
52 views

### Integral Measures: Identification

Problem Given a Borel space $\Omega$. Consider a Borel measure: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}:\quad\mu\geq0$$ Regard a Borel measure: ...
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32 views

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### On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form  0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
1answer
52 views

### Pointwise Convergence: No Diagonal Subsequence Exists?

Can anyone find a sequence of arbitrary functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, such that for each $n$, there is a ...
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39 views

### Haar Measure on Locally Compact monoids

I have been reading on Haar measure and we know that every locally compact Hausdorff group admits a Haar measure, is the same true for semigroups with identity $e$(monoid)? If not, is there a class of ...
1answer
82 views