Tagged Questions

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

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Using MCT twice to show the limit of an integral depending on $x$ and $n$

So I have $\displaystyle\lim_{n \to \infty} \int^{n^2}_0 e^{-x^2} n \sin\left(\frac{x}{n}\right) dx$. I'd like to apply the MCT but the trouble is there is a limit which also depends on $n$ So I ...
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What is the property $\mathfrak{F}$ in Fubini's theorem?

Notation) $\mathbf{x} = (x_1, \cdots, x_n)$, $\mathbf{y} = (y_1, \cdots, y_m)$, $I_1=\{\mathbf{x}: a_i\le x_i\le b_i, ~~i=1, \cdots, n\}$ $I_2=\{\mathbf{y}: c_j\le y_j\le d_j, ~~j=1, \cdots, m\}$, ...
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Contradiction to a Theorem for Lebesgue integrability proof

We know that $f$ is Lebesgue Integrable iff $|f|$ is Lebesgue Integrable. We have shown by contour integration that $\int^{\infty}_0 \frac{\sin(x)}{x}$ is $\frac{\pi}{2}$ - yet we can also show using ...
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Examples of functions where the Lebesgue integral as a measure is complete.

Let $f\in\mathcal{M}(\mathbb{R})$ non negative. For each $E\subset\mathbb{R}$ measurable we define $\mu_f(E)=\int_{E}f$. Prove (a) $\mu_f$ is a measure in $\mathcal{M}$ (b) Give an example of a ...
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Why is $\frac{1}{x}$ not Lebesgue integrable on $[0,1]$?

My teacher said (without explaining) that $\frac{1}{x}$ is not Lebesgue integrable on $[0,1]$? Could someone please explain why is this true?
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Lebesque integral is not injective?

I know that Lebesgue integral is not injective i.e. if $$\int\limits_{\Omega} f\, d\mu=\int\limits_{\Omega} g \,d\mu$$ then it is not necessary that $f=g$ on $\Omega$, But is there a simple ...
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if $\int{}f$ is finite, then $\int{}f$ exists?

My textbook said, If $\int_E f$ exists then, of course, $-\infty\le\int_E f\le+\infty$. If $\int_E f$ exists and is finite, we say that $f$ is Lebesgue integrable, or simple integrable, on $E$ and ...
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$f'$ Lebesgue-integrable

Let $f:[a,b]\to\mathbb R$ be differentiable and the derivative $f'$ bounded. How to show that $f'$ is Lebesgue-integrable on $[a,b]$ and $$\int_{[a,b]}f'd\mu=f(b)-f(a)$$ where $\mu$ denotes the ...
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Integral of convex function applied on a function

Let $f$ be an integrable function of $\mathcal{L}(\mathbb{C},\mathbb{R})$, measure Lebesgue. I want to prove that there exists an increasing convex function $H:\mathbb{R}^+\rightarrow\mathbb{R}^+$ ...
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Can someone solve my non-understandable process in proving a theorem?

Theorem. Let $E$ be a subset of $\mathbb{R}^n$. Then, if $p\gt0$, $\int_E|f-f_k|^p\to0$, and $\displaystyle\int_E|f_k|^p\le{}M$ for all $k$, then $\displaystyle\int_E|f|^p\le{}M$. For your ...
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Is right this application of Hadamard three-lines theorem for $\frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
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How to decide whether Lebesgue integral or Riemann integral?

Very often I feel very uncomfortable in dealing with integrals, since I am wondering whether the given integral is meant as a (improper) Riemann integral or Lebegue integral? For instance, the Gamma ...
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Non-regular measure can be represented by a regular measure

Let $X$ be a locally compact and Hausdorff space, and let $\mu$ be a positive measure on the Borel sets of $X$ (here $\mu$ is not necessarily regular). Then the linear map $L : C_c(X) \to \Bbb C$ ...
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