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Measure Theory Basics

stat.berkeley.edu · 2,533 words · saved by 1 readers

Measure theory is an area of mathematics concerned with measuring the “size” of subsets of a certain set. Soon after it was developed in the the early twentieth century, the great Soviet mathematician Kolmogorov realized it could be applied to give a rigorous grounding to probability theory, it was a major advance in understanding and resolving certain paradoxes in probability theory. David Aldous gives a nice discussion of this history. This is not a course on measure-theoretic probability and we will not rigorously develop the subject. However, it will be useful to draw on some of the basics of measure theory, to simplify our notation throughout the course and to clarify certain concepts around integration and conditioning. Homework 0 illustrates some of the interesting aspects of measure theory and why it is useful. Given a set 𝑋 , a measure 𝜇 is a certain kind of function mapping “nice enough” subsets 𝐴 ⊆ 𝑋 to non-negative numbers 𝜇 ( 𝐴 ) ∈ [ 0 , ∞ ] . Example 1 (Countin

Measure Theory Basics \[ \newcommand{\cB}{\mathcal{B}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cX}{\mathcal{X}} \newcommand{\EE}{\mathbb{E}} \newcommand{\PP}{\mathbb{P}} \newcommand{\RR}{\mathbb{R}} \newcommand{\td}{\,\textrm{d}} \] Measure theory: a rigorous grounding for probability Measure theory is an area of mathematics concerned with measuring the “size” of subsets of a certain set. Soon after it was developed in the the early twentieth century, the great Soviet mathematician Kolmogorov realized it could be applied to give a rigorous grounding to probabil

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