31  Probability

Definition 31.1 ($\mathbf{\text{Absolutely continuous}}$)
Consider a measure space $(\mathrm{\Omega },\mathcal{B})$, then a measure $\mu$ is said to be absolutely continuous with respect to $\nu$, namely $\mu <<\nu$, iff: $$\mu <<\nu ⟺\mu (B)=0⟹\nu (B)=0\mathrm{\forall }B\in \mathcal{B}$$

Definition 31.2 ($\mathbf{\text{Concentration}}$)
Consider a measure space $(\mathrm{\Omega },\mathcal{B})$, then a measure $\mu$ concentrates on $B\in \mathcal{B}$, if $\mu (B^c)=0$.

Definition 31.3 ($\mathbf{\text{Mutually singular}}$)
Consider a measure space $(\mathrm{\Omega },\mathcal{B})$, then two measures $\mu$ and $\nu$ are said to be mutually singular if for any disjoint set $A\cap B=\mathrm{\varnothing }$, $A,B\in \mathcal{B}$ we have that $\mu$ concentrates on $A$ and $\nu$ concentrates on $B$.

Definition 31.4 ($\sigma \mathbf{\text{-finite}}$)
Consider a measure space $(\mathrm{\Omega },\mathcal{B})$, then a measure $\mu$ is said to be $\sigma$-finite if exists a countable partition $B_1,B_2,\dots ,\subset \mathrm{\Omega }$ such that $\mathrm{\forall }i$ we have that $B_i\in \mathcal{B}$ and $\mu (B_i)<\mathrm{\infty }$. In other words, a measure is $\sigma$-finite, when we are able to divide the sample space in a countable partition of sets such that each one is in $\mathcal{B}$ and has finite measure. Note that this do not imply that the measure of $\mathrm{\Omega }$ is finite. In fact, consider for example the Lebesgue measure $m$ on $\mathbb{R}$, then we obtain $m(\mathbb{R})=\mathrm{\infty }$. However, since we can partition $\mathbb{R}$ in a countable series of intervals, each one with finite length, then the Lebesgue measure is $\sigma$-finite.