Questions tagged [measure-theory]

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

8,064 questions with no upvoted or accepted answers
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6
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597 views

Are Dynkin's $\pi-\lambda$ Theorem and the Monotone Class Theorem equivalent?

Let $X$ be a set. If $\mathcal{A}$ is a family of subsets of $X$ then let $\sigma(\mathcal{A})$ denote the $\sigma$-algebra generated by $\mathcal{A}$. Definition $1$ A $\pi$-system on $X$ to be a ...
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688 views

Lebesgue measurability: Rudin's vs Tao's

I'm a bit confused about Lebesgue measurability of a set. I saw two different forms of definition: (1) In Terence Tao's Introduction to Measure Theory, the definition is quite simple. First, the ...
6
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591 views

Define $\rho(f,g):=\int \frac{|f-g|}{1+|f+g|}d\mu.$. Show that $f_{n}\rightarrow f$ in measure $\Longrightarrow$ $\rho(f_{n},f)\rightarrow 0$.

Let $(X,\mathcal{M},\mu)$ be a measurable space, suppose $\mu(X)<\infty$. If $f$ and $g$ measurable functions on $X$, define $$\rho(f,g):=\int \frac{|f-g|}{1+|f-g|}d\mu.$$ Let $(f_{n})_{n\in\...
6
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2answers
151 views

Dividing a connected shape into equal-sized connected pieces

It's nearly pancake day here in the UK and I'll be making my kids pancakes. The last one in the batch is always a little oddly shaped, and I'd like to divide it into two pieces of equal area so they ...
6
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157 views

Why is the notation $\mu(dt)$ in integrals so widely used in general measure theory?

It seems to me rather ambiguous as it gives the impression of invariance by translation. Could it be an archaic notation that comes from Lebesgue measure (for which of course, it makes sense) ? It ...
6
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138 views

Sufficient conditions for separately measurable functions being jointly measurable.

Let $(X, \Sigma_X)$, $(Y, \Sigma_Y)$ and $(Z, \Sigma_Z)$ be measurable spaces and consider a mapping $f : X \times Y \to Z$. The following sufficient condition for the measurability of the sections is ...
6
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230 views

Linear separation theorem for closed convex sets of measures

Let $\mathcal P([0, 1])$ be the space of all probability measures on $[0, 1]$ endowed with the total variation metric. Let $P\subseteq \mathcal P([0, 1])$ be its closed convex subset, and $p'$ a ...
6
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61 views

Does the following condition suffice to imply convergence in distribution?

We know that convergence in distribution of random variables can be characterized as follows: Suppose $\{X_n\}$ and $X$ are defined on the same probability space. $X_n \stackrel{d}{\longrightarrow} X$ ...
6
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263 views

If $f \in L_p$ and $g \in L_\infty$, show that $fg \in L_p$.

If $f \in L_p, 1\le p\le \infty$, and $g \in L_\infty$, then the product $fg \in L_p$ and $\|fg\|_p \le \|f\|_p\|g\|_\infty$. I am still trying to show the first part, that $fg \in L_p$. What I have ...
6
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1answer
271 views

Image of a finite measure is a closed subset of $\mathbb R$

Let $(X,\Sigma,\mu)$ be a measurable space with $\mu(X)<\infty$. Prove that $\mu(\Sigma)$ is closed. I've been stumped with this for quite a while. I've tried every usual way of showing a set is ...
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1k views

If $f_n \to f$ and $g_n \to g$ in measure and $\mu$ is finite, then $f_n g_n \to fg$ in measure

This is Problem 3.1.5 in Cohn's Measure Theory, 2nd edition. Let $\mu$ be a measure on $(X, \mathcal A)$, and let $f, f_1,f_2, \ldots$ and $g,g_1,g_2,\ldots$ be real-valued $\mathcal A$-measureable ...
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727 views

Capacity vs measure of a set - intuitive understanding

There is a concept of measure of "largeness" of a set, called capacity. The intuition is, instead of physical largeness (measured by Hausdorff or Lebesgue measure), capacity measures how good a given ...
6
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447 views

Proof verification: $\frac{d(\nu_1\times \nu_2)}{d(\nu_1 \times \nu_2)}(x_1,x_2)=\frac{d\nu_1}{d\mu_1)}(x_1)\frac{d\nu_2}{d\mu_2}(x_2).$

This is exercise 3.12 from Folland's Real Analysis. It took me a long times to come up with a solution to this problem, and I'd appreciate it if anyone could verify if my answer is correct. For $j=1,...
6
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162 views

Theorem about the liminf and limsup

Its apparently referred to as Besicovich theorem (I checked all of his available papers but they only one that seems relevant is related to Hausdorf measure and the set E is in the plane) : for $E\...
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390 views

Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a ...
6
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255 views

Limit of $L_p$ norm as $ p \rightarrow 0$

I have reviewed Ayman Houreih's proof for the limit of the $L_p$ norm as $ p \rightarrow 0$ at "Scaled $L^p$ norm" and geometric mean. While I have found the outline of the proof very ...
6
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231 views

Anti-random reals

EDIT: This has now been crossposted at MO: https://mathoverflow.net/questions/219366/antirandom-reals. This is partially motivated by my question at mathoverflow: https://mathoverflow.net/questions/...
6
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367 views

Differentiation under the integral sign when derivative exists only almost everywhere

Regarding the Theorem 3 from here (or pdf ver.). Let $X$ be an open subset of $\mathbb{R}$, and $\Omega$ be a measure space. Suppose that a function $f\colon X\times\Omega\to \mathbb{R}$ satisfies ...
6
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327 views

Cutting a Banach-Tarski Cake

I was reading a cake-cutting problem here (not really related, so I won't link to it), and for some reason, this variation occurred to me. I have no idea whether this problem is even well-formed: ...
6
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203 views

Does a choice of measure on $\mathfrak{g}$ induce a measure on $G$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. One can put a (left) Haar measure $\mu$ on $G$ and a Lebesgue measure $\lambda$ on $\mathfrak{g}$ which are both unique up to constants. My ...
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170 views

Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure.

Let $(X,M, μ)$ be a measure space and $0 < p < q ≤ ∞$. Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure. My work: I proved the ...
6
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1answer
323 views

Proving that every set in the ring generated by all rectangles can be covered by a finite disjoint union of rectangles

Let $\mathcal{J}^n$ by the collection of all "rectangles" in $\mathbb{R}^n$, that is: $[[a,b))\in\mathcal{J}^n\iff [[a,b))=[a_1,b_1)\times[a_2,b_2)\times\cdots\times[a_n,b_n)$ where $a,b\in\mathbb{R}^...
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1k views

Question on Proof: when two measures agree on $\pi$-system, then they agree on generated $\sigma$-algbra

Let $\mu_1, \mu_2$ be measures on $\sigma(\mathcal P)$, where $\mathcal P$ is a $\pi$-system, and suppose they are $\sigma$-finite on $\mathcal P$. If $\mu_1$ and $\mu_2$ agree on $\mathcal P$, then ...
6
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672 views

Why does the union of all open null sets is itself a nullset for second countable space?

On the online Encyclopedia of mathematics, it is written "The existence of a countable base guarantees that the union of all open μ-null sets is itself a nullset." See: https://www.encyclopediaofmath....
6
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1answer
375 views

Definition and derivation of conditional expectation/probability

I read quite a few books introducing the notion of conditional probabilities/expectation by putting a formula out there coming from what they call "intuition". Can someone provide me a good measure ...
6
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174 views

Question about B. Host paper 'Nombres, normaux entropie, translations'

I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = \mathbb{R}/\mathbb{Z}$, $p > 1$ a integer, $D_n = \{kp^{-n}\colon 0 \leq k < p^n \}$...
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141 views

Is $\sigma$-finiteness really a necessary condition for this problem?

Question: Let $(X, \mathcal A, \mu)$ be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given any $\varepsilon$, there exists a $\delta >0$ such that $$\...
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303 views

Lebesgue density theorem for compact metric spaces.

Let $X$ be a compact metric space (with balls $B_{\varepsilon }(x)$), $\mu $ a Borel probability measure, and $A$ a Borel set with positive probability. Do we have that $\lim_{\varepsilon \...
6
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1answer
159 views

$u\in W^{1,p}(B)$ implies $u\in W^{1,p}(\partial B_t)$ for almost $t\in (0,1]$?

Suppose that $u\in W^{1,p}(B)$ where $B=B(0,1)\subset\mathbb{R}^N$ and $p\geq 1$. It was showed on this post (as an application of Fubini's theorem) that for almost $t\in (0,1]$ $$u_{|\partial B_t},(\...
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182 views

Measurability of a certain set in Falcolner's Geometry of Fractal Sets

On page 24 of Falcolner's The Geometry of Fractal Sets, Falcolner defines the set $F = \{ x \in E : \mathcal{H}^s(E \cap U) < \alpha$ diam$(U)^s$, for all convex sets $U$ containing $x$ such that $...
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2k views

“Simple” proof of Lebesgue's Differentiation Theorem for dimension 1?

Lebesgue Differentiation Theorem for $\mathbb{R}$: Let $f:[a,b]\to \mathbb{R}$ be intergable and $F(x)=\int_a^xf$. Then $F$ is differentiable almost everywhere in $[a,b]$ and $F'=f$ a.e. Is there a (...
6
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1answer
201 views

Baire sets of $X$ possess the required Cartesian product property

Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\;|\; E_{i}\; \text{is a Borel set in}\; X_{i}\; ,\; \text{for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in ...
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73 views

Shift operator on locally compact groups

Assume $f:G\rightarrow H$ is a measurable function between two locally compact abelian groups and let $T^h(f) = f\circ T^h$, where $T^h(x) = x-h$ (group operations in G and H are written additively). ...
6
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0answers
141 views

Haar measure existence using distributions

The book of Lieb and Loss proves the existence of Lebesgue measure in an unorthodox way as theorem 6.22, using the fact that ''positive distributions are measures''. My question is, whether it is ...
6
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1answer
336 views

Finding two functions (density) $g,f$ satisfying some conditions

Is there a clever way to find two density functions, $f$ and $g$, that satisfy the following conditions? $$\begin{align*} \int_{\infty}^{m}\int_{-\infty}^{\infty}f(w)f(w+z)\,dw\,dz&=\int_{\infty}...
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0answers
165 views

Convergence of integrals of Radon measures

Let $X$ be a locally compact Hausdorff space and let $\mu_n$ be a sequence of bounded variation Radon measures on $X$ such that $\int_X g \;d\mu_n \rightarrow \int_X g \;d\mu$ for each $g \in C_0 (X)$ ...
6
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1answer
2k views

Vitali Covering

Let $E$ be a set of finite outer measure and $F$ a collection of closed, bounded intervals that cover $E$ in the sense of Vitali. Show that there is a countable disjoint collection {${I_k}$}$_{k=1}^\...
6
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2answers
197 views

Measure theory question about sum of sequence of functions

Let $\left(f_n\right)_{n\ge1}$ be a sequence of measurable real valued functions. Prove that there exist a sequence of constants $c_n$ $>0$ such that $\sum_{n=1}^{\infty} c_nf_n $ converges for ...
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0answers
25 views

Is the collection $\mathcal{M}$ of $\mu$-measurable sets maximal so that $\mu|_{\mathcal{M}}$ is a measure?

Let $\mu:2^{X} \to [0, \infty]$ be an outer measure. The collection $\mathcal{M}$ of $\mu$-measurable sets are then defined as those sets $A$ satisfying $\mu(S)=\mu(S \cap A) + \mu(S \setminus A)$ for ...
5
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2answers
80 views

IF $\mu_n \rightarrow \mu$ Show that , $\sup _{A\in \mathbb{R}}|\mu_n -\mu |\rightarrow0$

Let $\mu_n , $ be probability measures on $( \mathbb{R}, \mathcal{R})$ with $n \geq 1$ with charachterstic functions ${\Phi}_n$. $\mu$ is also a probability measure with function $g$ Given that $|...
5
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1answer
119 views

Change of variable in Lebesgue integral

In my book it is shown how to compute the following Lebesgue integral: $$\int e^{-x^2} =\sqrt{\pi}$$ I want to show that this result is equivalent to $$\int e^{-x^2/2} =\sqrt{2\pi}$$ Up to now we ...
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0answers
113 views

Are jointly measurable adapted processes relative to natural filtration from right-continuous processes, progressively measurable?

Recently i'm studying the book of Stroock and Varadhan - "Multidimensional Diffusion Processes" and i'm trying to solve this exercise : For the first part i argue in the following way : "Because ...
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0answers
77 views

Regular conditional probability on Polish space and absolute continuity

Let $(\Omega,\mathcal F,\mathbb P)$ is a standard Borel space (i.e. $\Omega$ is Polish and $\mathcal F = \mathcal B(\Omega)$). Then $\mathcal F$ is separable and for every sub-sigma-algebra $\mathcal ...
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0answers
81 views

Compute the integral of $e^{-x}$

I am working on a problem in my past Qual. "Prove that $e^{-x}$ is Lebesgue integrable on $[0,\infty)$ and compute the integral." Here is my solution: $e^{-x}$ is continuous, hence measurable. We ...
5
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0answers
174 views

For an uncountable set $I$, is every Borel set of $2^I$ measurable?

Let $I$ be an uncountable set. Consider $2^I$ to be a measure space with the usual coin-tossing measure. That is, we define a measure on $2^I$ as follows. First we define $\mu(N_p) = 2^{-|p|}$, where ...
5
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0answers
443 views

What is the measure of $\int_{A}^B a_{\frac{x-A}{dx} } f(x) dx$?

Mathematicians like to ask covering various sets with open intervals and the answers to these riddles have strange tendencies to become strange lemma's or theorems Heine-Borel Theorem, lebesgue's ...
5
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0answers
72 views

Integral with respect to spectral measure

Let $A:D(A)\subset H \to H$ be a self-adjoint unbounded operator on complex Hilbertspace $H$ with corresponding spectral measure $E:\mathcal{B}(\mathbb{R})\to\mathcal{L}(H)$. I want to show that an ...
5
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0answers
97 views

Calculating the probability of a sequence having $n$ terms in a certain set.

Let $\Delta$ be the interval $[0,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),m)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $m$ is the Lebesgue ...
5
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0answers
45 views

Time derivative of a measure and convolution

Suppose $\mu_t : [0, + \infty) \to \mathcal{P}(\mathbb{R}^n)$ is a curve of probability measures on $\mathbb{R}^n$. For each $\nu \in \mathcal{P}(\mathbb{R}^n)$ I define $\nu \ast \rho$ as the ...
5
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0answers
74 views

Topology and $\sigma$-algebra on the space of probability measures

Let $(X,\mathbf{A})$ be a $\sigma$-algebra of sets, $\mathscr{M}(\mathbf{A})$ be the set of all $\mathbf{A}$-measurable functions $f\colon X\to [0,1]$, and $\mathscr{P}(\mathbf{A})$ be the set of all ...