# Tagged Questions

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### Integration of a continuous function under Lebesgue-Stieltjes measure space using simple functions

I am struggling to prove the following result using an approximating sequence of simple functions. Could anyone give me a clue? Under a Lebesgue-Stieltjes measure space ...
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### Calculate Radon-Nikodym derivative in a point when it is continuous in that point

I can't solve the following exercise, even if I find it quite intuitive. Let $\nu, \mu$ be Radon measures on a metric space $(X,d)$. Suppose that: 1) $w\in L^1(X,\mu), w\geq 0$ $\mu$ a.e.; 2) $w$ is ...
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### Continuity of a parametric integral (where the integrated function is discontinuous)

For all $t\in\mathbb{R}$ consider $$F(t):=\int_\mathbb{R}e^{-x^2/2}\log|t+e^x|\,dx \;.$$ I managed to show that $F(t)$ is well-defined and finite for every $t$. I would like to show that $F$ is ...
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### Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
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### prove $\exists$ a continuous function g s.t. $|g(x)-f(x)|<\epsilon$ for all x in a subset E of (a,b) and $\mu[(a,b)-E]<\delta$

I am having trouble starting with ths problem. Let f be a Lebesgue-integrable function over a bounded interval (a,b). Prove that for any $\epsilon >0$, $\delta >0$, there exists a continuous ...
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### Generalization of absolute continuity with $f(x) = x^a \sin(1/x^b)$

As a generalization of Prove that $x^\alpha \cdot\sin(1/x)$ is absolutely continuous on $(0,1)$ : Let $f : (0, 1] \to \mathbb{R}$ be the function denoted by $f(x) = x^a \sin(1/x^b)$. Determine for ...
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### Continuity of $L^1$ functions with respect to translation

Let $f\in L^1$, consider the map $t\mapsto f_t=f(x-t)$, then how can one show that $t\mapsto f_t$ is continuous? More explicitly one wants to show that $\lim_{h\to 0}|f_{t+h}-f_t|_{L^1}=0$. I tried to ...
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### Continuity and Integrability

Any help with this problem is appreciated. Consider the function $f(x) = \sum_{n=1}^\infty x n^{-\beta} e^{-nx}$. For what values of $\beta \in \mathbb{R}$ is $f$ continuous on $[0,\infty)$ and in ...
### {$\int_{[1/n,1]}f$} 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 ...
Luzin's theorem states that: let $f:[a,b]\rightarrow R$ be an a.e. finite function, $f$ is measurable iff $\forall \epsilon \geq 0: \exists \phi_\epsilon$ continuous on $[a,b]$ and \$\mu\{x: f(x)\neq ...