Chapter Summary

Chapter Summary

Key Points

• 1.

  The Lebesgue integral, which partitions the range of a function instead of the domain, generalizes the Riemann integral and provides the rigorous foundation for expectation of arbitrary random variables — discrete, continuous, mixed, or singular.

• 2.

  A $\sigma$-algebra $\mathcal{F}$ specifies which subsets of $\Omega$ are "measurable," a measure $\mu$ assigns non-negative values to them, and a measurable function (random variable) is one whose preimages are in $\mathcal{F}$.

• 3.

  Conditional expectation $\mathbb{E}[X \mid \mathcal{G}]$ is a $\mathcal{G}$-measurable random variable, not a number. It is the best $L^2$ approximation of $X$ using only the information in $\mathcal{G}$, and it satisfies the tower property, linearity, Jensen's inequality, and the "taking out what is known" rule.

• 4.

  Martingales are adapted sequences where $\mathbb{E}[X_{n+1} \mid \mathcal{F}_n] = X_n$ — modeling fair games. The martingale convergence theorem and the optional stopping theorem are powerful tools for proving convergence results and computing stopping-time expectations.

• 5.

  The Radon-Nikodym theorem states that $\nu \ll \mu$ implies the existence of a density $d\nu/d\mu$. This unifies PDFs ($dP_X/d\lambda$), PMFs ($dP_X/d\text{counting}$), and likelihood ratios ($dP_1/dP_0$) under a single framework, extending hypothesis testing and estimation to settings without classical densities.