> [!abstract] > Countably additive set functions $\mu:\mathcal{F} \to [0,\infty]$ are called **measures** when they are paired with a measurable space $(\Omega, \mathcal{F})$, assuming that $\mathcal{F}$ is a $\sigma$-algebra. A measure $\mu$ is: - **Finite** if $\mu(\Omega)<\infty$, so by additivity and non-negativity, $\mu(A)<\infty$ for all $A \in \mathcal{F}$. - **σ-Finite** if there is countable collection $(S_{n}) \subseteq \mathcal{F}$, where $\cup_{n}S_{n}=\Omega$, and $\mu(S_{n})<\infty$ for each $S_{n}$. - A **probability measure** if $\mu(\Omega)=1$, in which case the measure space $(\Omega, \mathcal{F}, \mu)$ is a **probability triple**. > [!math|{"type":"theorem","number":"","setAsNoteMathLink":false,"title":"Holder's Inequality","label":"holder-inequality"}] Theorem (Holder's Inequality).