Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
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Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance
I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement:
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How to prove limit of measurable functions is measurable
I need help to prove the following theorem
Suppose $f$ is the pointwise limit of a sequence of $f_n$, $n = 1, 2, \cdots$, where $f_n$ is a Borel measurable function on $X$. Then $f$ is Borel ...
user239780
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If a continuous function is positive on the rationals, is it positive almost everywhere?
I made up this question, but unable to solve it:
Let $f : \mathbb R \to \mathbb R$ be a continuous function such that $f(x) > 0$ for all $x \in \mathbb Q$. Is it necessary that $f(x) > 0$ ...
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To show that the set point distant by 1 of a compact set has Lebesgue measure $0$
Could any one tell me how to solve this one?
Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$
Show that $A$ has Lebesgue measure $0$.
Thank you!
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Showing that rationals have Lebesgue measure zero.
I have been looking at examples showing that the set of all rationals have Lebesgue measure zero. In examples, they always cover the rationals using an infinite number of open intervals, then compute ...
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Steinhaus theorem (sums version)
This is a question from Stromberg related to Steinhaus' Theorem:
If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval.
I can't quite see how to modify the Steinhaus ...
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What is the definition of a measurable set?
I have seen multiple definitions for what a measurable set is (all of which come together to form a sigma algebra). I was wondering if they are all equivalent and if not what situation would one be ...
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Explain densities to me please!
When it comes to integration on manifolds, I speak two languages. The first is of course the language of differential forms, which is something I am relatively well acquainted with.
The second ...
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Correspondences between Borel algebras and topological spaces
Though tangentially related to another post on MathOverflow (here), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...
Suppose $\...
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What is the Kolmogorov Extension Theorem good for?
The Kolmogorov Extension Theorem says, essentially, that one can get a process on $\mathbb{R}^T$ for $T$ being an arbitrary, non-empty index set, by specifying all finite dimensional distributions in ...
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Does there exist a continuous function $f: \Bbb R\to \Bbb R$ that is rational at (Lebesgue) almost every irrational, and irrational at every rational?
Does there exist a continuous function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ that is rational at (Lebesgue) almost every irrational, and irrational at every rational?
Some thoughts: for some $q\in ...
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The measurability of convex sets
How to prove the measurability of convex sets in $R^n$? I have seen a proof, but too long and not very intuitive. If you have seen any, please post it here.
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Convergence of random variables in probability but not almost surely.
Can somebody provide me with a sequence of (real-valued) functions, say on $[0,1]$ with the Lebesgue measure, such that the sequence converges in probability, or maybe in $\Vert \cdot \Vert _{L^2}$, ...
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Metric assuming the value infinity
If we instead define a metric as $d:X\times X \rightarrow [0,\infty]$, do we lose any nice properties of metric spaces?
The reason I ask is that I saw this theorem: Given a finite measure space $(X,\...
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Example of an algebra which is not a σ-algebra.
I have troubles with constructing an example of an algebra of sets which is not a σ-algebra. Could you please help me with this?
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