$\sigma$-algebras and measures

A $\sigma$-algebra $\mathcal{F}$ on $\Omega$ is a collection of subsets closed under countable unions and complements. A measure $\mu: \mathcal{F} \to [0,\infty]$ is $\sigma$-additive:

\[\mu\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \mu(A_i) \quad \text{for disjoint } A_i\]

A probability measure additionally satisfies $\mu(\Omega) = 1$.

The Lebesgue integral

For a non-negative measurable function $f$:

\[\int f\,d\mu = \sup\left\{\int s\,d\mu : s \text{ simple}, 0 \leq s \leq f\right\}\]

Dominated convergence: if $

f_n

\leq g$ with $\int g\,d\mu < \infty$, then $\int f_n \to \int f$.

Radon-Nikodym and change of measure

If $\nu \ll \mu$ (absolute continuity), there exists a density $f = d\nu/d\mu$ (Radon-Nikodym derivative) such that $\nu(A) = \int_A f\,d\mu$. In probability this is the likelihood ratio — central to change of measure in finance (risk-neutral pricing).