# Tagged Questions

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

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### Convex conjugate of average Fisher information measure

What is a possible convex conjugate of the function $\rho \mapsto \int (\nabla \log \rho(x))^2 \rho(x) dx$? (Suppose $\rho$ is a sufficiently integrable probability density function on a $d$-...
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### Determining Class of a general Borel measure

Let $(X, \mathcal{T})$ be a topological space, and $\Sigma = \Sigma(\mathcal{T})$ the $\sigma$-algebra of Borel sets (that is, the $\sigma$-algebra generated by $\mathcal{T}$). In Real Analysis and ...
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### Evaluating $\lim_{n\to\infty}\int_0^n(1-(x/n))^ne^{x/2}dx$

$$\mbox{How to compute}\quad \lim_{n \to \infty}\,\,\int_{0}^{n}\left(1 -{x \over n}\right)^{n} \,\mathrm{e}^{x/2}\,\,\mathrm{d}x\,\,\, ?.$$ No ideas how to start this one. I see that the limit of ...
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### Riesz measure in potential theory

I am studying Riesz measures associated to superharmonic funcions, following a book by Doob: Potential theory and its Probabilistic Counterpart. On page 51, the following theorem is introduced: If $u$...
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### $\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin y}{ny(1+n^2y^2)}ndy$ via DCT?

I'm looking to calculate these limits/integrals: $$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin (x/n)}{x(1+x^2)}dx$$ 2.$$\lim_{n\to\infty}\int_0^{\infty}\dfrac{\sin(x/n)}{(1+x/n)^n}dx$$ I posted ...
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### Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$ where $f\in L^{1}[0,1] \cap L^{2} [0,1]$

Let $f\in L^{1}[0,1] \cap L^{2} [0,1]$. Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$. I think the result would be $\left\|f\right\|_{1}$,but I don't know how to prove it.
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### Showing that every $\sigma$-algebra is a semiring (of sets)?

Let $X$ be a set. A semiring (of sets) is a collection $\cal{S}\subset$ ${{\cal{P}}}(X)$ such that $$\emptyset\in\cal{S}$$ $$S,T\implies S\cap T\in\cal{S},$$and for $S,T\in\cal{S}$ there is a finite ...
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### Why is compactness required for Brunn-Minkowski theorem?

Brunn-Minkowski theorem reads as follows: Consider two nonempty compact sets $A, B \subset \mathbb{R}^n$. Then the following inequality holds  [M(A+B)]^{\frac{1}{n}} \geq [M(A)]^{\frac{1}{n}} + [M(...
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### Limit of measures, two questions on limits of integrals

Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Call the limit $\mu(A)$. I ...
### What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?
Suppose $\epsilon \in (0, 1)$ and $m$ is Lebesgue measure. What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?