# Tagged Questions

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

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### Hellinger integral properties - proof of equivalence for infinite product measures

I'm trying to prove that: Let $(\mu_k)_{k=1}^{\infty}$ and $(\nu_k)_{k=1}^{\infty}$ be sequences of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Consider the product measures on ...
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### Integrability of a measurable function

Hi everyone: Suppose $(E_{n})$ is an increasing sequence of sets in $\mathbb{R}^{p}$ $(p\geq2)$ such that $\bigcup_{n}E_{n}= B$, a ball in $\mathbb{R}^{p}$. Suppose also that $f$ is a measurable ...
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### Carother's “certainly” proof about measurable sets

Carother's Real Analysis text has the following Theorem. Can someone check if my proof is correct? $(i \Rightarrow ii)$ Let $E$ be a measurable. Let $I_k$ be open intervals, such that m^*(E) ...
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### Why Are Some Sets Not Measurable?

I'm trying to understand why you can't evaluate a measure on generic sets (the ones in Banach-Tarski construction). That is, I want to know why when considering $m(X)$, we have to restrict our ...
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### Definition of Hausdorff Measure: example question

I am studying the Hausdorff measure and dimension, but I am struggling to understand the reason that the $n$-dimensional Hausdorff measure is zero for a set with Hausdorff dimension $<n$. The ...
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### Visualization of Fubini's Theorem

I understand that Fubini's Theorem is vital to evaluating double and triple integrals (via the equivalence of iterated integrals) especially in elementary multivariable calculus, and I know that it ...
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### Lebesgue measure of union of semi-open interval

Given $\mathbf{A} = \bigcup_{n\geq0}[n,n+ \frac{1}{2^n}[$ and the Lebesgue measure $\lambda$, find $\lambda(\mathbf{A})$. My solution: \begin{align} &\lambda\left(\bigcup_{n\geq0}[n, ...
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### When Borel functions and Baire functions are equal?

Suppose $X$ is compact metric space. Let $A$ be the smallest set of complex functions containing all continuous functions such that: If $f_n \in A$ are uniformly bounded and $f_n \to f$ pointwise ...
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### Integral of a function $f: \mathbb{N}\times\mathbb{N} \to\mathbb{R}$

Let $X = Y = \mathbb{N}$, $A = B = P(\mathbb{N})$, $\mu$ and $\nu$ counting measures on $(X, A)$ and $(Y, B)$. Define $f:X\times Y \to \mathbb{R}$ by $f(m,m) = 1$, $f(m+1,m) = -1$ and $f(m,n) = 0$ ...
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### Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0 < P(B) < 1$

Let $(\Omega, \mathbb{F}, P)$ be a probability space. Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0$ $<$ $P(B)$ $<$ $1$. My ...
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### Function of bounded variation and integration

Let f belong to $C[a,b]$. Show that there is a function g that is of bounded variation on [a,b] for which $\int_a^bfdg=||f||_{max}$ and TV(f)=1. This problem appears on page 162 of Royden's Real ...
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### If $\{f_n\}$ is Cauchy in measure, then there is a measurable function $f$, such that $\{f_n\}$ converges in measure to $f$

The theorem is from Real Analysis (Carothers). Let $\{f_n\}$ be a sequence of real valued measurable functions, all defined on a common measurable domain $D$. If $\{f_n\}$ is Cauchy in measure, then ...
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### We have sums, series and integrals. What's next?

We know how to sum or average a finite number of terms: sums. We know how to sum a countable infinite number ${\beth_0}$ of terms: series. We know how to sum ${\beth_1}$ terms: integrals. How to ...
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### Markov Inequality proof (measure theory)

I am trying to prove Markov's Inequality in measure theory as: Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be a non-negative function which satisfies $g(x)>0$ se $x>0$, and not descendant in ...
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### Characterization of support of positive regular Borel measures

Let $\mu$ be a positive Borel measure ona compact Hausdorff topological space. I am trying to prove the following: Show that $x \in support(\mu)$ if and only if $\int_X f d \mu >0$ for every ...
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### $x \in supp(\mu)$ iff $\int f d\mu >0$ for every $f \in C_c(X,[0,1])$ with $f(x)>0$

I'm reviewing for a real analysis midterm and have a question about this problem ($\mu$ is a Radon measure). I have two separate solutions to the "if" part, but have a question about each one. ...
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### Comparing Patrick Billingsley's Aniversary Edition to previous editions, and to Robert B. Ash's book.

I'm reading some of the reviews at amazon to the Anniversary edition of Billingsley's 'Probability and Measure', and several users state that the book is riddled with new typos, and plain errors, ...
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### Quotient of measurable functions

I need to show that if $f,g : X \to \mathbb{R}$ are measurables with respect to the $\sigma- algebra$ $S$ of $X$, and $g(x) \neq 0, \forall x \in X$, then $f/g: X \to \mathbb{R}$ is measurable. So ...
Let $X$ be a metric space. By definition, the sequence of Borel measures $\mu_n$ on $X$ converges weakly to a measure $\mu$, if for all bounded continuous functions $f:X\to\mathbb{R}$ we have ...