Questions tagged [measure-theory]
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Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
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Impossibility of probability measure on $2^{[0,1]}$
Let $\Omega = [0, 1]$ and define a set function on the subsets $(a, b] \subset \Omega$ as $P( (a, b] ) = b-a$ Prove that no extension of $P$ from the subsets where it is defined to the power set of $\... probability probability-theory measure-theory lebesgue-measure- 49
Simultanious convergence in $L^p$ and $L^q$
I have a question and unfortunately now idea yet, how to answer it. Suppose, $p$ and $q$ are numbers $>1$. Suppose $f_n$ are all in $L^p$ and $L^q$ (over some general measure) and $f_n$ converges ... measure-theory lebesgue-integral lp-spaces- 1,333
Measurable functions could be injective but not bijective?
I am trying to understand the concept of measurable functions. In several texts I found that if $(\Omega, \Sigma)$ is a measurable space, then $f: \Omega \to \mathbb{R}$ is measurable if and only if ... real-analysis measure-theory- 17
Failure of the Lebesgue differentiation theorem
Following wikipedia, we know that the Lebesgue differentiation theorem holds for quite general measures on separable metric spaces as long as they are finite dimensional, more explicitly if the ... measure-theory metric-spaces lebesgue-integral measurable-functions- 1,482
Can we determine the transition semigroup of a Markov process by its resolvent?
Let $(E,\mathcal E)$ be a measurable space, $$\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$$ be equipped with the supremum norm and $(\kappa_t)... probability-theory measure-theory stochastic-processes markov-process semigroup-of-operators- 12.8k
Weak convergence from convergence of integrals of polynomials
Let $\mu_n$ for $n\in \mathbb{N}$ and $\mu$ be probability measures on $\mathbb{C}^d$ with uniformly bounded support. Suppose that $$\int_{\mathbb{C}^d} f(z_1, \ldots, z_d) d\mu_n \to \int_{\mathbb{C}^... measure-theory reference-request moment-problem- 521
Please recommend a good textbook on measure theory, real analysis
I have read that G.B.Folland's real analysis. I like the contents that I covers because, like baby Rudin, I can study a lot of content fast. Nevertheless, I took a lot of time to understand since the ... real-analysis measure-theory reference-request book-recommendation- 21
Measure of null (or full) subsets of manifolds and absolute continuity of smooth measures
It is common to define null (resp. full) subsets of a $n$-dimensional manifold $M$ in the following way: $A \subseteq M$ is a null subset if its preimage under every coordinate chart is a null (resp. ... measure-theory differential-geometry- 107
Proove verification: $\int_{0}^{1}I_{\omega \notin \mathbb{Q}}(w) d\mathbb{P(\omega )}=1$ using Borel Cantelli lemma
I am just a student in probability theory and I am trying to understand by myself why: $\int_{0}^{1}I_{\omega \notin \mathbb{Q}}(w) d\mu(\omega )=1$ with $\mu (\omega )$ the Lebesgues measure. 1-To ... measure-theory solution-verification lebesgue-integral irrational-numbers rational-numbers- 314
Baby Rudin theorem 11.42
There are the definitions which we need for the proof of the theorem : There is the theorem: If ${f_n}$ is a Cauchy sequence in $\mathscr L^2(\mu)$ , then there exists a function $f$ $\in$ $\mathscr ... integration functional-analysis measure-theory multivariable-calculus cauchy-sequences- 395
Existence of Borel sigma Algebra
We can generate the borel sigma algebra by taking an intersection of all the sigma algebras containing the open sets. This collection is non empty because we can just take the powerset of the real ... measure-theory borel-sets- 1
If $(\kappa_t)$ is a Markov semigroup, is $t\mapsto\left\|\kappa_tf\right\|_\infty$ Borel measurable?
Let $(E,\mathcal E)$ be a measurable sapce, $$\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$$ be equipped with the surepmum norm and $(\kappa_t)... probability-theory measure-theory- 12.8k
Existence of "half-open interval parametrization" for metric measure spaces
Let $X$ be a Polish space, and let $\mu$ be a non-zero finite Borel measure on $X$ (the metric measure space $(X, \mu)$ is called a Lebesbue-Rokhlin space by Gromov in [1]). In the beginning of ... probability-theory measure-theory metric-spaces- 629
How can we show $\operatorname E\left[Y_\tau;t\ge\tau\mid\tau=s\right]=1_{\{\:t\:\ge\:s\:\}}\operatorname E\left[Y_s\right]$?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$; $(Y_t)_{t\ge0}$ be a real-valued $(\mathcal F_t)_{t\ge0}$-... probability-theory measure-theory stochastic-processes stopping-times- 12.8k
Representations of linear functionals on probability measures.
Let $A$ be a linear functional from $\mathcal{F}$ to $\mathbb{R}$. Where $\mathcal{F}$ is a set of finite signed measures on the the Borel sets in $\mathbb{R}^p$. What I want to know is whether there ... functional-analysis measure-theory topological-vector-spaces- 578
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