Questions tagged [measure-theory]

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

The modern notion of measure, developed in the late 19th century, is an extension of the notions of length, area or volume. A measure $\mu$ assigns numbers $\mu(A)$ to certain subsets $A$ of a given space. More specifically, a measure is a function from a $\sigma$-algebra to the extended real line (i.e. it may take infinite values). A $\sigma$-algebra is a collection of subsets of a set $X$, including $X$ itself and closed under complements and countable unions. A measure $\mu$ on a $\sigma$-algebra $\Sigma$ must satisfy the following properties:

  1. Nonnegativity: For every $A\in\Sigma$, $\mu(A)\ge0$.
  2. Null empty set: $\mu(\varnothing)=0$.
  3. Countable additivity: For every sequence $(A_n)_{n=1}^\infty$ of pairwise disjoint sets in $\Sigma$, $\mu\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\mu(A_n)$.

The notion of measure is a natural generalization of the following notions:

  1. Length of an interval
  2. Area of a plane figure
  3. Volume of a solid
  4. Amount of mass contained in a region
  5. Probability that an event from $A$ occurs, etc.

It originated in real analysis and is used now in many areas of mathematics, including geometry, probability theory, dynamical systems, functional analysis, etc.

Reference: Measure Theory

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Is this really a categorical approach to integration?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical Approach to Integration". The Abstract: "We…
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$L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample in the case the condition is not satisfied?
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Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$? I don't know where to start.
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The sum of an uncountable number of positive numbers

Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable). Proof: Let…
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Is it possible for a function to be in $L^p$ for only one $p$?

I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that if a function is in two $L^p$ spaces, (e.g. $p_1$…
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Construction of a Borel set with positive but not full measure in each interval

I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere. To be precise, if $\mu$ denotes Lebesgue measure, how would one construct a Borel set $A…
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What are Some Tricks to Remember Fatou's Lemma?

For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality $$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow \infty}(\int f_n \mathrm{d} \mu)$$ or alternatively (for…
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Does convergence in $L^p$ imply convergence almost everywhere?

If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$?
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If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$

Let $1\leq p < \infty$. Suppose that $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$. Why is it the case that $$\|f_k - f\|_{L^p} \to 0?$$ A…
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If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable

I am going over a tutorial in my real analysis course. There is an proof in which I don't understand some parts of it. The proof relates to the following proposition: ($S$ - infinite $\sigma$-algebra on $X$) $\implies $ $S$ is…
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Lebesgue measure theory vs differential forms?

I am currently reading various differential geometry books. From what I understand differential forms allow us to generalize calculus to manifolds and thus perform integration on manifolds. I gather that it is, in general, completely distinct from…
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Reference book on measure theory

I post this question with some personal specifications. I hope it does not overlap with old posted questions. Recently I strongly feel that I have to review the knowledge of measure theory for the sake of starting my thesis. I am not totally new…
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Understanding Borel sets

I'm studying Probability theory, but I can't fully understand what are Borel sets. In my understanding, an example would be if we have a line segment [0, 1], then a Borel set on this interval is a set of all intervals in [0, 1]. Am I wrong? I just…
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Integration of forms and integration on a measure space

In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus): the indefinite integral $\int f$ (also known as the…
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Set of continuity points of a real function

I have a question about subsets $$ A \subseteq \mathbb R $$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize this kind of sets? In a topological,measurable…
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