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Questions tagged [measurable-sets]

For questions about measurability, that is, whether a subset of a general space belongs to the σ-algebra, or about properties of measurable sets. Use this tag with (measure-theory), (real-analysis), (probability-theory) or (geometric-measure-theory).

Intuitively, a measurable set is a set which can be assigned a meaningful "size" (formally known as "measure"). The notion of measurability allows formal definition of length, and in , and probability of events in .

In formal settings, a subset $E$ of the general space $\Omega$ equipped with a $\sigma$-algebra ${\cal F}$ is called measurable if $E \in {\cal F}$.

Remark: Note that the general space $\Omega$ does not need to have, a priori, a topological structure.

Reference:

  1. Wikipedia
  2. Wolfram MathWorld
262 questions
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Convergence of Measurable Sets

Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of measurable subsets of $\mathbb{R}$. Suppose $A \subset \mathbb{R}$ such that $A_n \subset A$ for each $n \in \mathbb{N}$ and $m^{*}(A \setminus A_n) \rightarrow 0 $. Show that $A$ is lebesgue…
dazai osamu
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I have a set/venn diagram which I cannot solve.

I have two sets: |A| = 13 |B| = 26 |AUB| = 32 (U is the universal set) |U| = 50 Given the following information, label the Venn diagram, with the appropriate number of elements in each attribute. I want to use this equation -> |AUB| = |A| + |B| - |A…
Tariq Tayebi
  • 13
  • 2
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Set of numbers that are solutions for equations such as $ x = x + 1 $

Is there any number set that is used in that kind of equations without solution on the complex numbers?
Perch
  • 127
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Numerable union of open measurable sets

So I've come across this question: given $A_{1},A_{2},...\ $ Jordan measurable sets in $\mathbb{R}^{n}$, and given that $A=\cup_{n=1}^{\infty}{A_{n}}$ is bounded, is $A$ J-measurable?. Now, I can think of $\cup_{q\in[0,1]\cap\mathbb{Q}}{\{q\}}$ as…
valkyriebel
  • 188