Expectation

Introduction

Integration can be thought of as a function that takes a probability measure, a set, and a random variable

\[\begin{align*} &\mathcal{I} :: \mathcal{F} \to \mathcal{M} \to \{\Omega \to \mathcal{R} \} \to \mathcal{R} \cup \{ \pm \infty\}\\ &\mathcal{I} \ A \ \mathbb{P} \ X = \int _A X d\mathbb{P} \end{align*}\]

Lebesgue Integral

The lebesgue intergral is simply the following where \(\lambda\) denotes the lebesgue measure

\[\mathcal{I}_{\lambda}\]

Linearity

You have probably heard in various classes that integration is a linear function. What people mean by this is that the following higher-order function is linear

\[\mathcal{I} \ \mathcal{P} \ A \]

Working Across Probability Spaces

\[\int _A fd\mathbb{P} = \int_{f(A)} x d\mathbb{P}_f\]

To Do

is \(f(A)\) a measurable set??