Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
41,278 questions
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On the rate of convergence of $\log(\|f\|^p_p)$ as $p\rightarrow\infty$
Throughout this posting $(X,\mathcal{F},\mu)$ is a probability space and $f\in L_\infty(\mu)\setminus\{0\}$.
It is well known that $f\in L_p(\mu)$ for all $0<p<\infty$, $\phi(p)=\|f\|^p_p$ is ...
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Comparing hitting measure of two random walks on trees
I encounter a problem in random walks that I am not familiar with. I will appreciate it if anyone can give me a hint or references!
Let $T$ be a rooted trivalent tree. Denote $RW_1$ the random walk on ...
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Understanding a standard trick that $\mathbb{E}(T) = \mathbb{E}\left(\sum_{n=0}^\infty 1_{\{T > n\}}\right)$
I was working through essentially the question posed in here: If a stopping time, $T$, satisfies $P(T>k\alpha)\leq (1-\epsilon)^k$ then $E(T)<\infty$ regarding sufficient conditions when a ...
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Monotonicity and convexity of the $\log$ of the $1/r$ generalised mean.
I am currently working through a functional analysis textbook for summer reading, and I have become quite stuck on this question regarding the monotonicity and convexity of $\log M_r(a)$, where $a$ is ...
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Clarification on convergence in measure
I'm not exactly sure what to make of this. I am following Folland's second edition of Real Analysis, Modern Techniques and Their Applications. Suppose $(X,\mathcal{M},\mu)$ is a measure space. Let $|\...
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Are these three set functions logically consistent?
1. Definitions
Three functions are defined on subsets of the unit interval:
$$
\lambda, \nu, \kappa : \mathcal{P}([0,1]) \to [0,1]
$$
Let $M \subseteq [0,1]$. Consider the following collections:
$\...
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Looking for an article constructing a variant of the real numbers with infinitesimals and random sampling of subsets
I recall seeing a paper many years ago, which I’ve not been able to find since, that constructed an unusual number system which was designed to formalize some notions that are intuitive but ill-...
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Continuity of measures
In measure theory, standard textbooks typically emphasize the following continuity properties of measures:
Let $(\Omega, \mathscr{B}, \mu)$ be a measure space. Then:
(a) $\displaystyle A_k\in\...
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Williams Probability Theory with Martingales, asking clarification to measurability check on Chapter 9 "Conditional Expectation", page 83 "
I have not had to think about conditions for measurability in a few years, so I am bit rusty with nitty gritty details. Williams has defined the r.v. $Y$ on a probability space $(\Omega,\mathcal{F},\...
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Do integer translates $f(x+n)$ of an $L^1$ function converge to zero a.e.?” [closed]
Let $f\colon \mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue measurable function defined everywhere such that
$$
\int_{\mathbb{R}} |f| \, dm < \infty,
$$
where $m$ denotes the Lebesgue measure.
...
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Under what condition integrand is positive for all $(x_1,x_2)$. [closed]
Let $X=(X_1,X_2)$ be a bivariate random vector with support set $(0,s_1)\times(0,s_2)$ and join distribution function $F_X(x,y)$ with marginals distributions $F_{X_1}$ and $F_{X_2}$. Under what ...
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Second fundamental theorem of calculus for Henstock-Kurzweil integral
Let ${[a,b]}$ be a compact interval of positive length. We say that a function ${f: [a,b] \rightarrow {\bf R}}$ is Henstock-Kurzweil integrable with integral ${L \in {\bf R}}$ if for every ${\...
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Questions on product measures and Tonelli's theorem [closed]
I'm reading Sheldon Axler's MIRA book and have some thoughts that I want to know whether are correct based on the following definition and results:
Definition: Suppose $(X, \mathcal{S}, \mu)$ and $(Y,...
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Layer Cake Representation without Fubini (Exercise 1.6 of Lieb and Loss)
I would like a hint in the right direction for the following problem:
Lieb and Loss' Analysis defines the general Lebesgue integral as a special case of the layer cake representation: for a non-...
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Compact product measure has compact marginal distributions
The following is an offhand remark in Bogachev's Measure Theory, Volume $(2)$:
Let $(X,\mathscr{A}_X)$, $(Y,\mathscr{A}_Y)$ be measurable spaces and $\mu$ a probability measure on $(X\times Y,\...
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