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Added notation pages for major sections.

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Bokuan Li
2026-04-23 02:56:47 -04:00
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src/measure/index.tex
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\input{./measurable-maps/index.tex}
\input{./lebesgue-integral/index.tex}
\input{./bochner-integral/index.tex}
\input{./notation.tex}

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src/measure/notation.tex Normal file
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\chapter{Notations}
\label{chap:measure-notations}
\begin{tabular}{lll}
\textbf{Notation} & \textbf{Description} & \textbf{Source} \\
\hline
$\sigma(\mathcal{E})$ & $\sigma$-algebra generated by $\mathcal{E}$. & \autoref{definition:generated-sigma-algebra} \\
$\lambda(\mathcal{E})$ & $\lambda$-system generated by $\mathcal{E}$. & \autoref{definition:generated-lambda-system} \\
% ---- Measure Theory ----
$\mathcal{B}_X$ & Borel $\sigma$-algebra on $X$. & \autoref{definition:borel-sigma-algebra} \\
$\sigma(\{f_i \mid i \in I\})$ & $\sigma$-algebra generated by the maps $\{f_i\}$. & \autoref{definition:generated-sigma-algebra-function} \\
$\bigotimes_{i \in I} \mathcal{M}_i$ & Product $\sigma$-algebra. & \autoref{definition:product-sigma-algebra} \\
$\chi_E = \mathbf{1}_E$ & Indicator function of $E$. & \autoref{definition:indicator-function} \\
$\Sigma(X, \mathcal{M}; E)$ & Space of $E$-valued simple functions on $(X, \mathcal{M})$. & \autoref{definition:simple-function-standard-form} \\
$\Sigma^+(X, \mathcal{M})$ & Space of non-negative simple functions. & \autoref{definition:simple-function-scalar} \\
$\mathcal{L}^+(X, \mathcal{M})$ & Space of non-negative measurable functions. & \autoref{definition:measurable-non-negative} \\
$f_*\mu$ & Pushforward of $\mu$ by $f$. & \autoref{definition:pushforward-measure} \\
$\mu \otimes \nu$ & Product measure. & \autoref{definition:product-measure} \\
$|\mu|$ & Total variation measure of a signed/vector measure. & \autoref{definition:total-variation-signed}, \autoref{definition:total-variation-vector} \\
$\mu = \mu^+ - \mu^-$ & Jordan decomposition of a signed measure. & \autoref{theorem:jordan-decomposition} \\
$\mu \perp \nu$ & Mutual singularity. & \autoref{definition:mutually-singular} \\
$\nu \ll \mu$ & $\nu$ is absolutely continuous w.r.t. $\mu$. & \autoref{definition:absolutely-continuous} \\
$M(X, \mathcal{M}; E)$, & Space of finite $E$-valued measures. & \autoref{definition:vector-measure-finite-space} \\
$\|\mu\|_{\mathrm{var}}$ & Total variation of $\mu$. & \autoref{definition:vector-measure-finite-space} \\
$M_R(X; E)$ & Space of finite Radon $E$-valued measures on $X$. & \autoref{definition:space-radon-measures} \\
% ---- Lebesgue Integral ----
$\mathcal{L}^p(X, \mathcal{M}, \mu; E)$ & Space of $p$-integrable functions, without quotient. & \autoref{definition:lp-unequivalence} \\
$\|f\|_{L^p}$, & $L^p$ norm of $f$. & \autoref{definition:esssup} \\
$L^p(X, \mathcal{M}, \mu; E)$ & Space of $p$-integrable functions, modulo equality almost everywhere. & \autoref{definition:lp} \\
\end{tabular}