# Tagged Questions

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

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### Prove absolute continuity without Banach-Zarecki

Let $f$ be a real-valued continuous function of bounded variation on $[a,b]$. Suppose $f$ is absolutely continuous on $[a+\eta,b]$ for every $\eta\in(0,b-a)$. Show that $f$ is absolutely continuous on ...
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### How to calculate sigma algebra generated by random variables in stochastic process?

According to http://www.math.uah.edu/stat/processes/Stop.html, a stochastic process $X=\{X_t:t \in T\}$ is a stochastic process with state space $(S, \mathscr{S})$ defined on underlying probability ...
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### A Measure Theory problem-If $\int_{A_n}f(x)dx\rightarrow0$ then $\lambda(A_n)\rightarrow0$

This question was proposed as part of a test for PhD applicants but considered too hard and rejected. I tried unsucessfully to solve it for quite some time. For anyone wishing to try his luck.. ...
Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ two sub-sigma-algebras and $f$ a $\mathcal{B}$-measurable function. I want to show, that the subspace $$\... 2answers 22 views ### Natural filtration of martingales I don't quite understand what the natural filtration really is. Imagine e.g. a sequence of independent and identically distributed random N(0,1) variables. What is their natural filtration, and how ... 1answer 34 views ### Measure theory: upper bound for a particular set I have the following problem: consider A_1,...,A_N Borel set on [0,1] with measure greater than 1/2. For every a real number between 0 and \frac{1}{2}, consider the set E_a={x| x\in A_j ... 0answers 22 views ### Problem 1.3 from RCA Rudin Prove that if f is a real function on a measurable space X such that \{x : f(x) \geqslant r\} is measurable for every rational r, then f is measurable. Proof: For every \alpha\in \mathbb{R}... 0answers 25 views ### Situation where the conditions of Kolmogorov Consistency Theorem not hold [closed] I'm wondering what is a possible finite dimensional distribution that violates the two conditions in the Kolmogorov extension theorem. It's hard for me to imagine what distribution violates these ... 1answer 19 views ### \sigma-finite versus Locally Finite Measures Which implications are true (if any) for a measure \mu: \sigma- finite \implies locally finite locally finite \implies \sigma-finite My guess would be that both are false, but ... 1answer 35 views ### An example of outer measure. First a few definitions: 1.5.1: Definition. Suppose that \mu is a nonnegative set function on domain \mathcal{A} \subset 2^X. A set A is called \mu-measurable if for any \epsilon>0, ... 0answers 14 views ### Bayesian equation: need for priors As far as I understand, in the problem of Bayesian inference we have a random variable y describing data, which is distributed according to some parameter x via the known conditional distribution ... 1answer 43 views ### Is \iint \dfrac{1}{z} dxdy\neq 0? I am trying to solve an exercise and at some point I came accross the integral$$\iint_L \dfrac{1}{z} dxdy,$$(z=x+iy) where L\subset \mathbb{C} is a compact set with positive two-dimensional ... 0answers 34 views ### Is Bayesian Association mathematically rigorous? Introduction. This question is based on the Ph.D. thesis of B.T. Vo, which can be found in this website ("Papers" section). More specifically, in the introduction of the Ph.D. thesis, at page 8, there ... 0answers 14 views ### Function in indicator function and integral. I have the following expression$$\int 1_{ t^{-1}(A)}(x) e^{t(x)} d\mu(x) How do I express this integral in terms of $t(\mu)$? Specifically, what do I do about that indicator function which also ...
How can I show that the diagonal $D=\{(x,y)\in\mathbb R^2\vert x=y\}$ is a Lebesgue-nullset in $\mathbb R^2$ by utilizing the theorem of Tonelli? My solution so far, but it doesn't seem quite right: ...