⧼exchistory⧽
2 exercise(s) shown, 0 hidden
BBy Bot
May 08'24
[math]
\newcommand{\R}{\mathbb{R}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Rbar}{\overline{\mathbb{R}}}
\newcommand{\Bbar}{\overline{\mathcal{B}}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\p}{\mathbb{P}}
\newcommand{\one}{\mathds{1}}
\newcommand{\0}{\mathcal{O}}
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\newcommand{\mathds}{\mathbb}[/math]
Show that if [math]f,g\in\mathcal{L}^1(E,\mathcal{A},\mu)[/math] and [math]f\leq g[/math] a.e., then
[[math]]
\int fd\mu\leq \int gd\mu.
[[/math]]
BBy Bot
May 08'24
[math]
\newcommand{\R}{\mathbb{R}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Rbar}{\overline{\mathbb{R}}}
\newcommand{\Bbar}{\overline{\mathcal{B}}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\p}{\mathbb{P}}
\newcommand{\one}{\mathds{1}}
\newcommand{\0}{\mathcal{O}}
\newcommand{\mat}{\textnormal{Mat}}
\newcommand{\sign}{\textnormal{sign}}
\newcommand{\CP}{\mathcal{P}}
\newcommand{\CT}{\mathcal{T}}
\newcommand{\CY}{\mathcal{Y}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\mathds}{\mathbb}[/math]
Let [math]\mu[/math] be a measure on [math](\mathbb{R},\mathcal{B}(\mathbb{R}))[/math] such that for all [math]x\in\mathbb{R}[/math], [math]\mu(\{x\})=0[/math]. Moreover, let [math]\phi\in\mathcal{L}^1(\mathbb{R},\mathcal{B}(\mathbb{R}),\mu)[/math] such that
[[math]]
\int_\mathbb{R}\vert x\phi(x)\vert d\mu \lt \infty.
[[/math]]
Furthermore, for [math]u\in \mathbb{R}[/math], define
[[math]]
F(u)=\int_{\mathbb{R}}(u-x)^+\phi(x)d\mu.
[[/math]]
Show that [math]F[/math] is differentiable and that
[[math]]
F'(u)=\int_{(-\infty,u]}\phi(x)d\mu.
[[/math]]