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Let $(X, \mathcal X, \mu)$ and $(Y, \mathcal Y, \nu)$ be $\sigma$-finite measure spaces. Let $\overline{\mathbb R} := \mathbb R \cup \{\pm \infty\}$. $f:X \to \overline{\mathbb R}$ is called $\mu$-si …

Let $d \in \mathbb N^*,p \in [1, \infty]$ and $T>0$. Let $$ F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0}), t \mapsto F_t $$ be measurable. I would like to ask if there is a measurable function $G …

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the …

First, we prove that for all $f \in S(X\times Y)$ and $\varepsilon >0$. There is $f_\varepsilon \in S(X\times Y)$ such that $g$ satisfies $(*)$ and that $\lambda (A) \le \varepsilon$ where $A := \{f …

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the …

Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, and $(E, | \cdot |)$ a Banach space. $S (X)$ the space of $\mu$-simple functions from $X$ to $E$. $\m …

Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue measu …

Inspired by two answers in this thread, I'm able to drop the assumption on the integrability of $f$, i.e., Theorem Let $p \in [1, \infty)$ and $f \in L^0 (Z)$ such that $$ f(x, \cdot) \in L^p(Y) \qua …

Let $\varphi: \mathbb{R}^d \to \mathbb{R}$ be a convex function. The subdifferential of $f$ at $x$ is defined as $$ \partial \varphi (x) := \{z \in \mathbb{R}^d : \varphi(y) \geq \varphi(x) + \langle …

I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please sav …