Measure Theory

Wed, 11 Nov 2026 13:30:00 +1100

This page sits between Real Analysis (metric spaces, completeness, uniform convergence) and Probability (which it puts on a rigorous footing). It condenses my notes for UNSW MATH5825 Measure, Integration and Probability (Term 3, 2026), which follow T. Tao’s An Introduction to Measure Theory for the analysis and then turn to probability.

The spine, in one paragraph: the Riemann integral cannot handle limits, so we rebuild “length” from countable covers (Lebesgue outer measure), keep only the sets on which it is additive (Carathéodory), and integrate by approximating from below with simple functions. Three convergence theorems (monotone, Fatou, dominated) are the payoff. Everything then abstracts to an arbitrary \(\sigma\)-algebra with a measure; products of measure spaces give Fubini and Tonelli; the integrable functions form complete normed spaces \(L^p\). A probability space is a measure space of total mass one, a random variable is a measurable function, and expectation is the integral — so the laws of large numbers, the central limit theorem, conditional expectation and martingales are all theorems about integrals.

Lebesgue on Aayush Bajaj's Augmenting Infrastructure

Measure Theory