Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
40,085
questions
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Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?
I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of $P$...
- 7,137
57
votes
4
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Is composition of measurable functions measurable?
We know that if $ f: E \to \mathbb{R} $ is a Lebesgue-measurable function and $ g: \mathbb{R} \to \mathbb{R} $ is a continuous function, then $ g \circ f $ is Lebesgue-measurable. Can one replace the ...
- 593
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votes
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answer
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Lebesgue measurable set that is not a Borel measurable set
exact duplicate of Lebesgue measurable but not Borel measurable
BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck...
In short: Is there a Lebesgue ...
- 2,085
56
votes
1
answer
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Are vague convergence and weak convergence of measures both weak* convergence?
For quite a long time, I have been confused about the definitions of weak convergence and vague convergence of measures among other modes of convergence that root from functional analysis, mainly due ...
- 47.7k
55
votes
7
answers
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False beliefs about Lebesgue measure on $\mathbb{R}$
I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
user365
54
votes
3
answers
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General Lebesgue Dominated Convergence Theorem
In Royden (4th edition), it says one can prove the General Lebesgue Dominated Convergence Theorem by simply replacing $g-f_n$ and $g+f_n$ with $g_n-f_n$ and $g_n+f_n$. I proceeded to do this, but I ...
- 6,534
54
votes
3
answers
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Preimage of generated $\sigma$-algebra
For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. ...
- 5,546
52
votes
3
answers
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Example where union of increasing sigma algebras is not a sigma algebra
If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra?
It seems closed under complement since for all $x$...
- 2,489
51
votes
5
answers
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Intuitive interpretation of limsup and liminf of sequences of sets?
What is an intuitive interpretation of the 'events'
$$\limsup A_n:=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}A_k$$
and
$$\liminf A_n:=\bigcup_{n=0}^{\infty}\bigcap_{k=n}^{\infty}A_k$$
when $A_n$ are ...
- 2,111
51
votes
3
answers
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On the equality case of the Hölder and Minkowski inequalities
I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8.
Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the ...
- 10.5k
50
votes
8
answers
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views
Why do we restrict the definition of Lebesgue Integrability?
The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.)
Why is it we ...
- 2,924
50
votes
3
answers
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Vitali-type set with given outer measure
Is it possible to construct a non-measurable set in $[0,1]$ of a given outer measure $x \in [0,1]$? This will probably require the axiom of choice. Does anyone have a suggestion?
Edit: I forgot to ...
- 14.7k
49
votes
3
answers
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What is the difference between outer measure and Lebesgue measure?
What is the difference between outer measure and Lebesgue measure?
We know that there are sets which are not Lebesgue measurable, whereas we know that outer measure is defined for any subset of $\...
- 611
49
votes
4
answers
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Infinite product of measurable spaces
Suppose there is a family (can be
infinite) of measurable spaces. What
are the usual ways to define a sigma
algebra on their Cartesian product?
There is one way in the context of
defining product ...
- 47.7k
48
votes
6
answers
7k
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What is the intuition behind Chebyshev's Inequality in Measure Theory
Chebyshev's Inequality Let $f$ be a nonnegative measurable function on $E .$ Then for any $\lambda>0$,
$$
m\{x \in E \mid f(x) \geq \lambda\} \leq \frac{1}{\lambda} \cdot \int_{E} f.
$$
What ...
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