# Tagged Questions

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### Convergence of Integrands and Integrals

Suppose $E \subset \mathbb{R}$ is compact. Is it possible to find a sequence of positive continuous functions $f_n: E \to \mathbb{R}$ such that for every $x \in E$ we have $$f_n(x) \to f(x)$$ for some ...
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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 ...
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### 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 ...
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### Finding a limit of an integral

I am trying to find the following limit. Let $X = [0,\infty)$ and $\mathbb B$ denote the Borel subsets in $[0,\infty)$, $\lambda$ the Lebesgue measure. Let $f_n : [0, \infty) \to \mathbb R$ be given ...
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### Limit outside compactly supported range converges to zero

Suppose $f\in C^\infty(\mathbb{R})$ is compactly supported on $[-N,N]$, and $1\leq p<\infty$, and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)\textrm dx=1$. Define ...
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### Supremum over compactly supported range converges to zero

Suppose $f\in C^\infty(\mathbb{R})$ is compactly supported on $[-N,N]$, and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ It follows ...
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### Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
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### Limit of sequence of integrals

$$\lim_{n\rightarrow \infty} \int _0^\infty \frac{(1-e^{-x})^n}{1+x^2}dx.$$ this is less then $$\lim_{n\rightarrow \infty} \int _0^\infty \frac{1}{1+x^2}dx.$$ so the limit should exist by DCT. But ...
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### Showing a Lebesgue integral exists, while another doesn't.

Consider $$f_p(x)=x^p \exp\left(-x^8\sin^2x\right)$$ I have to show that $f_2\in\mathscr L(0,+\infty)$ whilst $f_3\notin \mathscr L(0,+\infty)$. Now, I am looking at the case $p=2$. The problematic ...
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### For $f$ periodic, $g\to 0$ the integral of $fg$ converges (under some more conditions)

Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions such that: $f$ is periodic, with finitely many zeros in a period The average value of $f$ on a period is $0$ $g$ is monotonic decreasing and ...
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### Construction of a sequence of simple functions converging pointwise to a given function

Q1: How to construct a sequence of $\{f_n\}$ of simple function for a function $f$ such that $f_n\to f$ converges pointwise? Q2: If $f$ is measurable is $f_n$ also measurable for each $n$? Q2 is by ...
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### The General Lebesgue Integral

For a measurable function, $f$, on $[1, \infty)$ which is bounded on bounded sets, define $a_n = \int_n^{n+1} f$ for each natural number $n$. Is it true that $f$ is integrable over $[1, \infty)$ if ...
### {$\int_{[1/n,1]}f$} to converge and yet $f$ is not $L$-integrable over $[0,1]$
Let $f$ be a function on $[0,1]$ and continuous on $(0,1]$. I want to find a function $f$ s.t. {$\int_{[1/n,1]}f$} converges and yet $f$ is not $L$-integrable over $[0,1]$. My attempts: I've found ...
I have problem with the exercise that follows. Let $(z_m)_m \in R^n$ so that $\Vert z_m \Vert \rightarrow \infty$ when $m\to \infty$. Let $f:R^n \rightarrow [-\infty;+\infty]$ integrable. Show ...