# Questions tagged [measure-theory]

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

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### 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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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
### 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 ...