# Tagged Questions

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### Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
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### Are the measurable subsets closed under the symmetric distance metric?

Define the following pseudo-metric on the set of measurable subsets of $R$: $$D(A,B)=\operatorname{Length}((A\setminus B) \cup (B\setminus A)),$$ i.e., the distance between $A$ and $B$ is ...
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### Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
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### limit of the integrations of a sequence of integrable functions

Let $(f_n)^\infty_{n=1}$ be a sequence of Lebesgue integrable functions on $[0,1]$ such that $f_n$ converges to $f$ almost everywhere in $[0,1]$. Suppose further (a). ...
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### can $L^p$ norm convergence and pointwise monotonic imply pointwise convergence?

Let $(f_n)_{n=1}^\infty$ be a sequence of measurable function such that $\lim_{n\to\infty}||f_n-f||_p=0$. If for any $x\in \Omega$, $\{f_{n}(x)\}_{n=1}^\infty$ is a monotonic sequence, can we deduce ...
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### Uniform convergence almost everywhere characterization

I am being faced with a non-trivial characterization of uniform convergence almost everywhere. Suppose $(X,S,\mu)$ is a non-necessarily finite measure space. Take a sequence of measurable functions, ...
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### Does convergence of integrals imply a.e. convergence of functions?

Let $\{u_n\}_{n\in\mathbb{N}}\subseteq L^{\infty}([0,T],U)$ be a sequence of measurable functions, where $U=[0,\overline u]$ is a compact set, and suppose that $\lambda(t,du)$ is a probability kernel ...
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### Convergence of sums using Hilbert space techniques [duplicate]

Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_nb_n < \infty$ for any sequence $b_n$ satisfying $\sum_{n=1}^{\infty}b_n^2 < \infty$. Prove that ...
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### Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
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### Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
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### Is this set measurable? (Set of points where a sequence converges)

Let $M$ be a manifold. Suppose that $u_n:M \to \mathbb{R}$ are measurable and we have $u_n(s) \to u$ a.e. in $M$. Does it follow that the set $A=\{s \in M : u_n(s) \to u(s)\}$ and $A^c$ are ...
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### Convergence in measure of integrable functions implies limit is integrable?

I'm going through my handwritten notes for my upcoming exam (so not homework) and the above was stated but not proven in class. The full statement is a little different, but the above part is the only ...
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### weak convergence implies point-wise convergence?

If we have a bounded sequence $\{f_n\} \in L^p[a,b]$ that converges weakly to $f$ does this mean that the converges is also pointwise?? thank you.
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### If for a.e $t \in (0,T)$, $u_n(t,x) \to u(t,x)$ for a.e. $x \in \Omega$,is $u_n \to u$ a.e. in $(0,T)\times\Omega$?

If for almost all $t\in (0,T)$, we have $$u_n(t,x) \to u(t,x) \quad\text{a.e. x \in \Omega}$$ does this mean that $$u_n \to u \quad\text{a.e. in (0,T)\times\Omega}?$$ Here $\Omega$ is an open ...
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### Properties convergence of random variables

I want to understand the concepts of convergence of random variables better and therefore I wanted to find out how certain concepts that are straightforward for canonical convergence behave in this ...
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### Why is $s-\lim_{n\to\infty} f_n = f\Longleftrightarrow~\forall g\in L_{\mu}^{\infty}: \lim_{n\to\infty} f_n g\, d\mu=\int fg\, d\mu$?

We had some different definitions concerning convergence. And I am a bit confused about that. First of all I give you some preparing definitions and then the definitions I mean. Let $E$ be a Banach ...
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### Convergence almost everywhere implies convergence in measure, the proof thereof

Let $(E, \mathcal{E}, \mu)$ be a measure space, and $(f_n)_{n\in\mathbb{N}}$ and $f$ be measurable functions $(E, \mathcal{E}, \mu)\longrightarrow (\mathbb{R}, \mathcal{B})$. The first part of Theorem ...
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### An unknown function

I run into a function: $1_{[-n, n]^r}$. I guess this function equals 1 whenever x falls into $[-n, n]^r$. Am I right? I met this function in an analysis paper which deals with measure and density of ...
### Why is $g_n:=\inf\{f_n,f_{n+1},…\}$ integrable if $f_n$ are?
This question is motivated by the proof of the Fatou's lemma. My text defines $g_n:=\inf\{f_n,f_{n+1},\ldots\}$ and states that it's Lebesgue integrable (each $f_n$ is). We proved that point-wise ...