# Questions tagged [lebesgue-integral]

For questions about integration, where the theory is based on measures. It is almost always used together with the tag [measure-theory], and its aim is to specify questions about integrals, not only properties of the measure.

4,548 questions
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### Is the Lebesgue integral equal to the Lebesgue measure of the enclosed region?

Let $u:\mathbb{R}\to[0,+\infty]$ be Lebesgue integrable, and let $$A:=\{(x,y)\in\mathbb{R}^2 \; | \; 0\leqslant y\leqslant u(x)\}.$$ Thus $A$ is the set of all points enclosed between the graph of ...
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### Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$.

Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$. Since the question is asking for a sequence of continuous ...
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### Proof $\frac{\sin x}{\sqrt{x}}$ is not Lebesgue integrable

I am trying to prove that the function $\frac{\sin}{\sqrt{x}}$ is not Lebesgue integrable on $[0, \infty]$. The proof I have seen seems to use a comparison test: \begin{align*} \int_0^{\infty} \left \...
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### Is the dominated convergence theorem applicable whenever “THIS” theoem is applicable?

THIS theorem: Let $I =[a,b]$ be a closed and bounded interval and $\forall n\in \mathbb{N}$, $f_n:I \to \mathbb{R}$ be Riemann integrable on $I$. If the sequence $(f_n)$ converges uniformly to a ...
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### Evaluate $\lim_{n \to \infty}\int_{0}^{1}\frac{n+\cos^n(e^x)}{4n+x^4} dx$

Evaluate $\lim_{n \to \infty}\int_{0}^{1}\frac{n+\cos^n(e^x)}{4n+x^4} dx$ Attempt: We define $f_n(x)=\frac{n+\cos^n(e^x)}{4n+x^4}$ on the domain $(0,1)$. This sequence of functions $(f_n)$ converges ...
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### Motion of a particle defined via Lebesgue integral

Suppose $x_0 =0$. A particle moves as follows: $$x_t = \int_0^t a(s) ds$$ where $a: \mathbb{R}_+ \to \{-1,0,1\}$ is a measurable function. Suppose, I have that $a_s = 1$ if $x_s = 0$. I want to ...
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### Let $(f_n)$ be a sequence in $M^+(\mathbb{R})$ proove that $\displaystyle{\int \sum_{n=1}^{\infty}f_n = \sum_{n=1}^{\infty}\int f_n}$

How to prove the following exercise: Let $(f_n)$ be a sequence in $M^+(\mathbb{R})$ proove that $\displaystyle{\int \sum_{n=1}^{\infty}f_n = \sum_{n=1}^{\infty}\int f_n}$ it reminds me the ...
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### Give an example of strict inequality in Fatou's Lemma

The fatou's Lemma says Let $\left\{ f_n, n = 1,2,...\right\}$ be a sequence of non-negative measurable functions. Then $$\liminf \int f_n \geq \int \liminf f_n$$ Some hint? I think this Showing ...
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### Prove that $F\in L^{1}(\mathbb{R})$.

Let $f\in L^{1}(\mathbb{R})$ and $\forall x\in\mathbb{R},$ let $F(x)=\displaystyle\int_x^{x+1} f(t)\ dt$. Prove that $F\in L^{1}(\mathbb{R})$. Hint: Use Tonelli. Attempt: I indeed used Tonelli's ...
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### Does $\displaystyle\int_\mathbb{R} f_n\ dm\to \displaystyle\int_\mathbb{R} f\ dm$?

Let $f_n$ be a sequence of measurable functions on $\mathbb{R}$ converging a.e. to $f$. If $0\leq f_n\leq f$ a.e. Does it follow that \$\displaystyle\int_\mathbb{R} f_n\ dm\to\displaystyle\int_\mathbb{...