\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}

