# 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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### triple integral $\iiint e^{-x^2-y^2-z^2+xy+yz+xz} \,dx\,dy \,dz$

I need to solve this triple integral $$\iiint e^{-x^2-y^2-z^2+xy+yz+xz} \,dx\,dy \,dz$$ where $V$ is all $\Bbb R^3$ I've spent on this task a few hours, ok firstly ...
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### If $f_n \to g$ in $L_2$, and that $f_n = \mathbb{E}(f \mid \mathcal{F}_n)$, show if $f$ continuous, then $g(x)=f(x)$ for all $x$ with probability 1?

Let $\Omega = [0,1]$, and let $\mathcal{F}$ be the Borel $\sigma$ field of $[0,1]$. Also let $\mathbb{P}$ denote the Lebesgue measure on $\Omega$ ($\mathbb{P}(\Omega) = 1$). Let $\mathcal{F}_n$ ...
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### How to tell if a function is in $L^1$ or in $L^2$?

Given a function, what properties do I need to satisfy to tell whether a function is in $L^{1}(R)$ and/or $L^{2}(R)$ For example, $\frac{sin(x)}{|x|^{\frac{3}{2}}}$. Do I just need to test if the ...
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### construction of the gamma function

I have some questions regarding the construction of the gamma function below. First, why is $|t^{z-1}|=t^{\Re z-1}$? Next, why is $|f_z(t)|\le C_z e^{-t/2}$ for $t\ge 1$, and how can the precise value ...
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### $f=\sum_1^\infty n^{-1}(-1)^n\chi_{(n,n+1]}$ is not Lebesgue integrable but improper Riemann integrable

Show that $f=\sum_1^\infty n^{-1}(-1)^n\chi_{(n,n+1]}$ is not Lebesgue integrable but $\lim_{b\to \infty} \int_0^b f(x) dx$ (the limit of the Riemann integral) exists. I'm stuck on how to show this. ...
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### The integral $\int f \mu(dt)$ versus $\int f \mu(t) dt$

In the theory of continuous-time branching processes there is a quantity called the Malthusian parameter $\alpha \in \mathbb{R}$ defined by: $$\int_0^{\infty} e^{-\alpha t} \mu(dt) = 1$$ The ...
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### $0\le f_n\le g_n\le h_n$ $f_n$ and $h_n$ are Lebesgue integrable then $g_n$ is also.

I was trying to solve the following problem but unfortunately I'm unable to solve it. Please help me. Let $f_n,g_n,h_n$ be Lebesgue integrable functions with $0\le f_n\le g_n\le h_n$ for all $n$. ...
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### Approximation of Integrable function

Let $f$ be an integrable function over $E$. Then given $ε > 0$, there is a simple function $φ$, step function $ψ$, and a continuous function $g$ supported on a finite measure set such that 􏰁 􏰁 􏰁 ...
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### Compare measures of two sets

I am struggling with probably something very trivial. Consider a dynamical system whose evolution equation is given by: $x^+ = f(x)$, where $x,x^+\in R^n$ and $f(x)\in R^n$ is piecewise continuous ...
I am stuck with the final line on the image, why does $f_n \to f$ (which also increases), mean that $\cup_n A_n = \Omega$?