# Tagged Questions

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

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### Union of sets as the union of disjoint sets - Does the proof $\forall n\in \mathbb{N}$ implies the proof for infinity?

I managed to prove that: $$\displaystyle\bigcup_{i=1}^n A_i=A_1\cup(A_1^c\cap A_2)\cup(A_1^c\cap A_2^c\cap A_3)\cup\dots\cup(A_1^c\cap\dots\cap A_{n-1}^c\cap A_n)$$ for $\forall n \in\mathbb{N}$. ...
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### Relationship between measure theory and real analysis

Does measure theory generalize real analysis to abstract spaces? For example, you can now talk about convergence even on unordered fields.
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### Assumptions involving product spaces

Suppose a random variable $X$ is distributed in $\mathbb{R}^{n}$ and we are given that $X' = (X_{1}', X_{2}')$ for $X_{i}$ distributed on $\mathbb{R}^{n_{i}}$. In general, what assumptions can I make ...
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I have real-valued functions $\{f_n\},f$ on a subset $X\subset \mathbb R^n$ that are equicontinuous and I have Borel measures $\{\mu_n\},\mu$. I have that For each fixed $m$, $\int f_m d\mu_n\to\int ... 1answer 49 views ### What do we mean when we say that a function$f$takes the value$ \infty $? What do we mean when we say that a function$f$takes the value$ \infty $? In measure theory it is common to let mappings take values in the extended real number system. But still it doesn't make ... 1answer 60 views ### What are boundary effects in measure-theory? I often read the term "boundary effects" which seem to be the reason to look at the interior$A^°$and closure$\bar{A}$of Borel subsets$A$separately. What is so special about the boundary and what ... 3answers 74 views ###$f_n \to 0 a.e.$and$\lim \int f_n d\mu =0$but$\sup_n f_n$is not in$L^1$Give an example of a finite measure space$(X,M,\mu)$and a sequence of functions$f_n:X \to[0, \infty)$such that$f_n \to 0a.e.$and$\lim \int f_n d\mu=0$but$\sup_n f_n$is not in$L^1$I ... 1answer 74 views ### uncountable Lebesgue-null set of$[0,1]$such that$1_N$is not Riemann integrable Give examples of the following and justify 1) uncountable Lebesgue-null set$N$of$[0,1]$such that$1_N$is not Riemann integrable on$[0,1]$2) uncountable Lebesgue-null set$N$of$[0,1]$such ... 1answer 102 views ### Weak convergence on L^p Let Let$X=[0,1]$with the Lebesgue measure, find a sequence$\{f_n\}$of measurable functions$f_n:X \rightarrow{ \mathbb{R} } $such that:$f_n(x)\rightarrow{0}$almost everywhere$x∈[0,1]f_n$... 2answers 66 views ### Show that random walk is a random variable I am working on this question. Suppose$\{X_n, n \ge 1\}$are random variable on the probability space$(\Omega, \mathcal{B},P)and define the induced random walk by \begin{align*} S_0=0, \, ... 1answer 81 views ### Showing countable additivitiy of Lebesgue measure The following is taken from the classic Probability and Measure by Patrick Billingsley, Theorem 2.2 (page 26 in the 3rd edition). I have a question on his proof, but I give the necessary defintions to ... 1answer 214 views ### Convergence in Total Variation Implies Convergence in Distribution SupposeX,Y$are random variables. We define the total variation distance of random variables to be$d(X,Y)= \inf \{P(|X′−Y′|>0): X′,Y′$are couplings of$ X,Y$respectively$\}$. Does ... 0answers 82 views ### monotone convergence of$f_n$plus weak convergence of$\mu_n$implies convergence? I would like to know if somebody is aware of some result that looks like the following. Let us consider the space$C_b(X)$of continuous bounded function over a measurable space$X\$. Suppose that: ...
I sometimes read statements such as The integral $$\int_0^{\infty} dx \, \frac{\sin x}{x}$$ does not exist as a Lebesgue integral, because it is not absolutely convergent. But according to my ...