# Tagged Questions

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

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### a proposition in the construction of the Borel measures on the real line

In the construction of the Borel measures on the real line, the following proposition is used in Folland's Real Analysis: Here is my question: If one replaces $(a_j,b_j]$ with $[a_j,b_j)$, ...
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### What is the motivation behind the arbitrary union topological axiom?

1. Why is the arbitrary union axiom in the definition of topology necessary? 2. Why is it useful? Why might we expect ("intuitively") that it should be useful? 3. What is the (historical) ...
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### Prove $S_k$ is a stopping based on $A$ being previsible

Probability with Martingales: It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ ...
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### Almost surely interpretation in conditional probabilities

Consider the random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, P)$. Suppose $Y$ is a discrete random variable with support $\mathcal{Y}\subset \mathbb{R}$. Suppose $X$...
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### Existence of a Lebesgue measurable set

The following is from Carother's Real Analysis: Suppose that $E$ is Lebesgue measurable with $m(E)=1$. Show that there is a measurable set $F\subset E: m(F)=1/2$. Carothers offers a hint which ...
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### scotts theorem and other representation theorems for order aggreeing qualitative quantitative probability measures

Scotts theorem and other theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, finite cancellation axioms ...
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### Can Lebesgue Dominated Convergence always be used?

Suppose I want to find the derivative $$\frac{d}{dx}\int f(x,y) dy.$$ I want to know under what condition it would be equal to $$\int \frac{d}{dx}f(x,y) dy.$$ Of course, if I can find a suitable ...
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### Interchanging Expectation and Derivative

Suppose I have a random function, $f(x)(\omega)$. And that for fixed $\omega$, we have the derivative $g(x)(\omega)=\frac{d}{dx}f(x)(\omega)$. For a fixed $x$, I can find the expectation $E(f(x))$. ...
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### Real Analysis Folland Proposition 1.13

Proposition 1.13 - If $\mu_0$ is a premeasure on $\mathcal{A}$ and $\mu^*$ is defined by (1.12) then a.) $\mu^*|\mathcal{A} = \mu_0$ b.) every set in $\mathcal{A}$ is $\mu^*$-measurable ...
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### Any borel measure on a locally compact, sigma-compact, Hausdorff space is regular?

on the book "measure and integration theory" by Heinz Bauer , theorem 29.12, the same proof works for a sigma-compact space instead of a locally compact space with countable basis(which is in ...
Recently, out of curiosity, I looked up the list of questions for Princeton generals, and one caught my attention: Can you construct a measurable set on the interval $[0; 1]$ such that its ...