24 Notations
| Notation | Description | Source |
$\sigma(\mathcal{E})$ | $\sigma$-algebra generated by $\mathcal{E}$. | Definition 17.1.6 |
| $\lambda(\mathcal{E})$ | $\lambda$-system generated by $\mathcal{E}$. | Definition 17.3.3 |
| $\mathcal{B}_{X}$ | Borel $\sigma$-algebra on $X$. | Definition 17.2.1 |
| $\sigma(\{f_{i} \mid i \in I\})$ | $\sigma$-algebra generated by the maps $\{f_{i}\}$. | Definition 21.1.6 |
| $\bigotimes_{i \in I}\mathcal{M}_{i}$ | Product $\sigma$-algebra. | Definition 21.2.1 |
| $\chi_{E} = \mathbf{1}_{E}$ | Indicator function of $E$. | Definition 21.4.1 |
| $\Sigma(X, \mathcal{M}; E)$ | Space of $E$-valued simple functions on $(X, \mathcal{M})$. | Definition 21.4.3 |
| $\Sigma^{+}(X, \mathcal{M})$ | Space of non-negative simple functions. | Definition 22.1.1 |
| $\mathcal{L}^{+}(X, \mathcal{M})$ | Space of non-negative measurable functions. | Definition 22.2.1 |
| $f_{*}\mu$ | Pushforward of $\mu$ by $f$. | Definition 18.1.8 |
| $\mu \otimes \nu$ | Product measure. | Definition 18.8.1 |
| $|\mu|$ | Total variation measure of a signed/vector measure. | Definition 19.5.2, Definition 19.5.1 |
| $\mu = \mu^{+} - \mu^{-}$ | Jordan decomposition of a signed measure. | Theorem 19.1.8 |
| $\mu \perp \nu$ | Mutual singularity. | Definition 19.4.1 |
| $\nu \ll \mu$ | $\nu$ is absolutely continuous w.r.t. $\mu$. | Definition 19.3.1 |
| $M(X, \mathcal{M}; E)$, | Space of finite $E$-valued measures. | Definition 19.5.4 |
| $\|\mu\|_{\mathrm{var}}$ | Total variation of $\mu$. | Definition 19.5.4 |
| $M_{R}(X; E)$ | Space of finite Radon $E$-valued measures on $X$. | Definition 20.3.3 |
| $\mathcal{L}^{p}(X, \mathcal{M}, \mu; E)$ | Space of $p$-integrable functions, without quotient. | Definition 14.1.1 |
| $\|f\|_{L^p}$, | $L^{p}$ norm of $f$. | Definition 14.1.2 |
| $L^{p}(X, \mathcal{M}, \mu; E)$ | Space of $p$-integrable functions, modulo equality almost everywhere. | Definition 14.1.7 |