# Tagged Questions

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

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### Integral of $f'$ where $f$ is continuous on $[a,b]$ and differentiable over $(a,b)$.

There is a problem which states that if $f$ is a function continuous on $[a,b]$ and differentiable almost everywhere on $(a,b)$ whose $|\text{Diff}_\frac{1}{n} f| \leq g$ almost everywhere on $[a,b]$ ...
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### Counterexample for “Filtration of stopping time equals filtration generated by stopped process”

I am working in a discrete setting. Consider any stochastic process $(X_n)_{n\in\mathbb N}$ with its natural filtration $(\mathcal F_n)_{n\in\mathbb N}$ and a stopping time $\tau$. We know that ...
I want to solve this problem: If $f,g:X\to \overline{\mathbb R}$ are measurable and $c$ is any extended real value the function $$h(x)=\begin{cases}c& \text{if}\ f(x)+g(x)\ \text{is undefined}\\ ... 1answer 29 views ### What is the measure zero of uncountable set . Recently I was reading Methods of Real Analysis by Goldberg and had the following question. 7.1 Corollary: Every countable subset of \mathbb{R} has measure zero. How can we describe the ... 0answers 29 views ### The relationship between random variables, distribution functions and probability measures Given a probability space (\Omega,\mathcal{F},P), and a random variable X\colon\Omega\to\Bbb{R}, we can associate with it its distribution function F\colon \Bbb{R}\to[0,1] defined as ... 3answers 48 views ### Real Analysis, Folland Problem 1.3.11 Measures Background information - Let X be a set well equipped with a \sigma-algebra M. A measure on M (or on (X,M) or on X if M is understood) is a function \mu: M \rightarrow [0,\infty] such ... 0answers 15 views ### Modulus of Integral defines a measure? Say you have an integrable function f: \mathbb{R}^n \to \mathbb{R} which does change sign. Does the set function$$m(A):=\left| \int_A f \text{d} x\right|$$define a either a measure or a ... 1answer 21 views ### On the interpretation of the limsup of a sequence of events In the context of understanding Borel-Cantelli lemmas, I have come across the expression for a sequence of events \{E_n\}:$$\bigcap_{n=1}^\infty \bigcup_{k\geq n}^\infty E_k$$following Wikipedia ... 0answers 19 views ### Measurability and integrability of set and function My textbook said: Let E\subset\mathbb{R}^n, let G be an open set, and let |\cdot|_e denote outer measure. if \exists{}G s.t. E\subset{}G and |G-E|_e\lt\varepsilon for an any given ... 1answer 25 views ### f_n = (\frac{1}{n})\chi_{[n, +\infty)}. Find \lim \int f_n d\lambda. Let X = \mathbb R, \textbf{X} = \textbf{B} and \lambda the Lebesgue measure on \textbf{X}. I have the following: f_n = (\frac{1}{n})\chi_{[n, +\infty)}. I need to find the following: ... 1answer 34 views ### Sequence of Events - Basic understanding I have an idea of the meaning of a sample space, and the events included in a sigma algebra. However, I am stuck in the definition of a sequence of events. My difficulty is in the fact that in a ... 0answers 39 views ### Find the Lebesgue measure of the following sets. Find the Lebesgue measure of the following sets: i) A=(\cup_{n=1}^\infty [2^n, 2^n + \frac{1}{2^n})) \ \mathbb{Z} ii) B=(\cup_{n=1}^\infty (n^n, n^n + \frac{1}{2^n})) \cap \mathbb{Q}. For ... 0answers 12 views ### Given an event field, is there a random variable generating it? [duplicate] In probability space (\mathsf{\Omega},\mathcal{F},\mathrm{P}), for any event field \mathcal{G}\subset\mathcal{F}, there always exists a random variable X, such that \sigma(X)=\mathcal{G}? Is ... 0answers 20 views ### Conditional Distributions vs. Stochastic Processes Is the concept of a version of a stochastic process related to the concept of a version of a conditional distribution? And is a regular version of a stochastic process somehow the same thing as the ... 0answers 48 views ### Sigma-algebra on \Omega = \{1,2,3,4,5\} generated by \epsilon = \{\{1,2,4,5\},\{2,3\}\} Let \Omega = \{1,2,3,4,5\}. Let \epsilon = \{\{1,2,4,5\},\{2,3\}\}. Find \sigma (\epsilon), generated by (\epsilon), and justify answer. Could someone please give me some direction and ... 2answers 25 views ### Let f_n = (\frac{1}{n})\chi_{[0,n]} and f = 0. Show that (f_n) converges uniformly to f. Let f_n = (\frac{1}{n})\chi_{[0,n]} and f = 0. Show that (f_n) converges uniformly to f. I have never done an example of convergence of sequences that have characteristic (indicator) ... 2answers 37 views ### Real Analysis, Folland Problem 1.3.14 [duplicate] If \mu is semifinite measure and \mu(E) = \infty, for any C > 0 there exists F\subset E with C < \mu(F) < \infty. Attempted proof - Suppose E\in M with \mu(E) = \infty then ... 1answer 38 views ### Can the unit interval be the disjoint union of countably many “super-dense” parts? I'm curious about this question in the case where f is not necessarily measurable. I think what it comes down to is this: Is there an \varepsilon<1 and a partition of [0,1] in countably ... 1answer 55 views ### Decomposition of complex Radon measures Suppose you have a complex Radon measure \mu, treated as a distribution. Then does every such Radon measure admit a decomposition of the form \mu = \sum_{n=1}^\infty c_n \delta(x-\tau_n) + \hat f ... 0answers 24 views ### Is there a there a non intersecting mapping to unit square. Is there a way to go from the fat cantor set to a half unit square in a non intersecting way using Hilberts curve? How would I go about constructing a non intersecting space filling curve of non zero ... 1answer 48 views ### Is every set of measure zero countable? I know it is true that every countable set has measure zero, but is the converse true. Is it true that every set of measure zero is countable? 1answer 22 views ### Exercise 10.N of The elements of integration and Lebesgue measure Bartle's book If a_{mn}\ge 0 for m,n\in\mathbb{N}, then \sum_{m=1}^{\infty} \sum_{n=1}^{\infty}a_{mn}=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}(\le +\infty). 0answers 15 views ### Prove that \phi:\mathbb R^{p+q}\to\mathbb R^{m+n}:(x,y)\mapsto (f(x),g(y)) is measurable. If I have two measurable functions f:\mathbb R^p\to\mathbb R^m and f:\mathbb R^q\to\mathbb R^n, how can I prove that$$\phi:\mathbb R^{p+q}\to\mathbb R^{m+n}:(x,y)\mapsto (f(x),g(y)) is ...
The definition in Bogachev's book goes as follows: Now let $\mathcal{A}$ be a $\sigma$ algebra and let $\mu$ be a finite countably additive measure. 1.12.7 Definition. The set $A\in \mathcal{A}$ ...