30  Probability

Definition 30.1 (\(\color{magenta}{\textbf{Absolutely continuous}}\))
Consider a measure space \((\Omega, \mathcal{B})\), then a measure \(\mu\) is said to be absolutely continuous with respect to \(\nu\), namely \(\mu << \nu\), iff: \[ \mu << \nu \iff \mu(B) = 0 \implies \nu(B) = 0 \; \; \forall B \in \mathcal{B} \]

Definition 30.2 (\(\color{magenta}{\textbf{Concentration}}\))
Consider a measure space \((\Omega, \mathcal{B})\), then a measure \(\mu\) concentrates on \(B \in \mathcal{B}\), if \(\mu(B^c) = 0\).

Definition 30.3 (\(\color{magenta}{\textbf{Mutually singular}}\))
Consider a measure space \((\Omega, \mathcal{B})\), then two measures \(\mu\) and \(\nu\) are said to be mutually singular if for any disjoint set \(A \cap B = \emptyset\), \(A,B \in \mathcal{B}\) we have that \(\mu\) concentrates on \(A\) and \(\nu\) concentrates on \(B\).

Definition 30.4 (\(\color{magenta}{\sigma\textbf{-finite}}\))
Consider a measure space \((\Omega, \mathcal{B})\), then a measure \(\mu\) is said to be \(\sigma\)-finite if exists a countable partition \(B_1, B_2, \dots, \subset \Omega\) such that \(\forall i\) we have that \(B_i \in \mathcal{B}\) and \(\mu(B_i) < \infty\). 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 \(\Omega\) is finite. In fact, consider for example the Lebesgue measure \(m\) on \(\mathbb{R}\), then we obtain \(m(\mathbb{R})= \infty\). 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.