Tagged Questions

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

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Sigma algebra - motivation in measure theory

Taken from the Motivation section of sigma-algebra article: A measure on $X$ is a function that assigns a non-negative real number to subsets of $X$; this can be thought of as making precise a ...
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Approximating measurable sets of infinite measure by open sets

Following a question posted here: Approximating measures by open sets and compact sets. I wanted to ask, if I am given a measurable set $E\subseteq \mathbb{R}$ s.t. $m(E)=\infty$, then how can I find ...
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Difference between $A+B$ and $A \cup B$

Basic set theory: $A$ and $B$ are two sets. I assume that $A+B$ isn't the same as $A\cup B$. I know what $A\cup B$ is but what is $A+B$? The context: I need to show that if A and B are open, then A+B ...
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Showing the integral of a function is finite almost everywhere

Suppose$\ E \subset \mathbb R$ is closed. Let$\ d(y) = \inf \{|x-y| : x \in E \}$ and let $\ M(x) = \int_0^1\frac{d^a(y)}{|x-y|^{(1+a)}} dy$ , for some arbitrary constant $a$. Show that $\ M(x)$ is ...
Suppose $f$ is a nonnegative measurable function on a measure space $(X,M,\mu)$ satisfying $\int f d\mu < \infty$. Show that for every $\epsilon > 0$ there exists a $\delta > 0$ such that ...