Articles of lebesgue integral

Mollifiers: Derivative

This thread is meant as lemma for: Semigroups & Generators: Entire Elements: Construction Given a smooth mollifier: $\varphi\in\mathcal{L}(\mathbb{R}): \varphi’\in\mathcal{L}(\mathbb{R})$ Do the derivatives exist in the sense: $$f\in\mathcal{C}(\mathbb{R}):\quad\int_{-\infty}^{\infty}\frac{1}{h}\left\{\varphi(\hat{x}+h)-\varphi(\hat{x})\right\}f(\tfrac{1}{n}\hat{x})\mathrm{d}\hat{x}\to\int_{-\infty}^{\infty}\varphi'(\hat{x})f(\tfrac{1}{n}\hat{x})\mathrm{d}\hat{x}\quad(\|f\|_\infty<\infty)$$ (Note the unbounded domain of integration.)

version of the dominated convergence theorem where the almost-everywhere convergence is used

Let $f\in \mathcal{L}^0(S,\mathcal{S},\mu)$ be a function State and prove a version of the dominated convergence theorem where the almost-everywhere convergence is used. Is it necessary for all $\{f_n \}_{n∈\mathbb N}$ to be dominated by $g$ for all $x\in S$, or only almost everywhere? I don’t even have a direction. What do I need to show? […]

Show that the Fubini Tonelli theorem does not work for this function

Problem Statement: Let $X = Y = R$ and let $B$ be the Borel $\sigma$-algebra. Define $$ f(x,y) = \left\{ {\begin{array}{*{20}{c}} 1&{x \ge 0{\text{ and }}x \le y < x + 1}\\ { – 1,}&{x \ge 0{\text{ and }}x + 1 \le y < x + 2}\\ {0,}&{{\text{ else }}} \end{array}} \right.$$ Show that $\int […]

Lebesgue Spaces and Integration by parts

Suppose there exists a Lebesgue Space, $L_1$ and functions functions $\phi$, $\phi’$, $f$, and $f’$ functions where $$\phi, \phi’ \in L_1$$ By rule of integration by parts, $$uv|_a^b = \int_a^b udv + \int_a^b vdu$$ Let $$ u = \phi, du= \phi’$$ $$ v = f, dv = f’$$ Are there any properties of Lebesgue functions […]

Is $\sin(1/x)$ Lebesgue integrable on $(0,1]$?

Is the function $f(x)=\sin(1/x)$ Lebesgue integrable on $(0,1]$? I know that, as $f$ is continuous on the set, it is a measurable function. However, I’m stumped on how to go on. A nudge in the right direction would be greatly appreciated.

Mollifiers: Asymptotic Convergence vs. Mean Convergence

Problem Does asymptotic convergence imply mean convergence: $$\varphi\in\mathcal{L}_\text{loc}(\mathbb{R}_+):\quad\varphi(T)\stackrel{T\to\infty}{\to}\varphi_\infty\implies\frac{1}{T}\int_0^T\varphi(s)\mathrm{d}s\stackrel{T\to\infty}{\to}\varphi_\infty$$ Remark Three important classes fall under local integrability: $\mathcal{L}(\mathbb{R}_+),\mathcal{C}(\mathbb{R}_+),\mathcal{B}(\mathbb{R}_+)\subseteq\mathcal{L}_\text{loc}(\mathbb{R}_+)$

An equality concerning the Lebesgue integral

Let $f:X\mapsto[0,+\infty)$ be a non-negative measurable function defined on the space $X$, endowed with the complete $\sigma$-additive, $\sigma$-finite, measure $\mu$ defined on the $\sigma$-algebra of the measurable subsets of $X$. I have read that the following equality holds for the Lebesgue integral: $$\int_X f d\mu = \int_{[0,+\infty)} \mu(\{x\in X: f(x)>t\}) d\mu_t$$where $\mu_t$ is the usual […]

Show that for $|f_n| \le g_n$ $\forall n$: $\lim_{n\to \infty} {\int_E g_n } = \int_E g \Rightarrow \lim_{n\to \infty} {\int_E f_n } = \int_E f$

Let $(f_n)_{n \in \Bbb N}$ be a series of measurable functions on E, that converges almost everywhere pointwise towards $f$. Let $(g_n)_{n \in \Bbb N}$ be a series of on $E$ integrable functions that converge almost everywhere on $E$ pointwise towards $g$. Also, $|f_n| \le g_n$ $\forall n \in \Bbb N$. I have to show […]

Measurable functions on product measures

Let $ (X,\mu) $ be a measure space, and consider $X \times X$ with the product measure $\mu \times \mu $. Consider two functions $f$ and $g$ defined on $X \times X$ such that: $f$ is measurable. For a.e. x, the function $y \to g(x,y)$ is measurable. The function $\int g(x,y) dy$ is a measurable […]

Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{}$ almost everywhere.

Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{[0,1]}$ almost everywhere.$\textbf(Make\ sure\ that\ your\ proof\ is\ completely\ rigorous)$. I don’t know, which property to use. (It is not allowed to show it with $\epsilon-\delta-criterion$, our last topics were: Lp-Spaces, Radon-Nikodym Theorem, Riesz Representation Theorem,Lipschitz-Functions, Product measures, Fubini Theorem) but i can’t find […]