How big is a set? For a line segment the answer is its length; for a blob in the plane, its area. But what is the "length" of the rational numbers scattered along a line, or the total size of a fractal dust? Measure theory is the rigorous answer — a single, watertight framework for assigning a size to sets far too wild for ordinary geometry, and for building an integral that can handle them.

The prize is the Lebesgue integral, which succeeds where the school (Riemann) integral fails: it integrates ferociously discontinuous functions, and its limit theorems — swapping limits and integrals almost for free — are the engine of modern probability, Fourier analysis and functional analysis. It is also, at last, the true foundation of probability itself.

The big idea: measure first, integrate second

One thread runs through everything here. Riemann chops up the x-axis; Lebesgue chops up the y-axis and then asks "how big is the set of x landing in this height band?" — a question of measure. So the whole subject is built in order: decide which sets are measurable (a σ-algebra), assign them sizes (a measure), and only then define the integral as a sum over height bands. Get the measure right and the integral, and its miraculous convergence theorems, follow.

The shape of the journey

This course moves in two stages.

• Stage A — Measure & Lebesgue integration. From "why we need measure" to σ-algebras, the Lebesgue integral, the convergence theorems and the great structural results.
• Stage B — Measure-theoretic probability. The pay-off: probability rebuilt on measure theory, giving rigorous random variables, laws of large numbers and the central limit theorem.

Stage A — Measure & Lebesgue integration

1. The Need for Measure
2. Σ-Algebras
3. Measures and Measure Spaces
4. The Lebesgue Outer Measure
5. Measurable Functions
6. The Lebesgue Integral
7. The Monotone and Dominated Convergence Theorems
8. Fatou's Lemma
9. Lᵖ Spaces
10. The Radon–Nikodym Theorem
11. Fubini's Theorem

Stage B — Measure-theoretic probability

1. Probability Spaces
2. Random Variables as Measurable Functions
3. Expectation as an Integral
4. Independence and Product Measures
5. Modes of Convergence
6. The Laws of Large Numbers
7. Characteristic Functions and the Central Limit Theorem

Let's get started

We begin by breaking the school integral on purpose — meeting a function so jagged that Riemann simply gives up. That failure is exactly the doorway into measure theory.

Let's get started → The Need for Measure