A measure-theoretic probability course

Probability, rebuilt from scratch

Where counting runs out: how a handful of corner cases forced probability to grow σ-algebras, Lebesgue measure, and all the rest of the machinery.

You already know how to compute probabilities. Dice, cards, urns, Bayes' rule, the binomial and the normal — Probability 101 got you this far, and it works. Until it doesn't. Pick a random real number in [0, 1]. Run a coin forever. Ask for the probability of a set you cannot even describe. Condition on an event of probability zero. Watch a sample average refuse to converge.

Each of these is a corner case where the 101 framework doesn't just give the wrong answer — it cannot even pose the question. This course shows how those failures, one by one, forced mathematicians to rebuild probability on measure theory. We keep the proofs. We never leave the examples behind.

§The full syllabus

0

Motivation: Where Probability 101 Breaks

The course begins at the scene of the crime. We isolate exactly what the 101 framework assumes, and exhibit real problems where each assumption dies.

1

Sample Spaces and σ-Algebras

Events are sets, but not every set is an event. We build the machinery that decides which questions a probability model can even hear.

2

Probability Measures

The axioms, taken seriously. Continuity of probability, atoms, discrete spaces done honestly, and the monsters that finitely additive probability allows.

3

Lebesgue Measure

Length, upgraded. Outer measure, Carathéodory's construction, the sets that fail, and the famous pathological objects.

4

Random Variables and Measurable Maps

A random variable is a function with a constraint so quiet you'd miss it — and so important everything downstream depends on it.

5

Expectation and the Lebesgue Integral

Where the Riemann integral fails and Lebesgue's wins, and why expectation is an integral at all.

6

Product Spaces and Fubini

Independence as product measure, integrating in two orders, and the tail events that Borel–Cantelli governs.

7

Convergence Theorems

The payoff: the laws 101 promised you, proved with the machinery the earlier chapters built.

8

Conditional Expectation and Beyond

The end of the beginning: conditioning on σ-algebras, martingales, and where the road goes next.

How this course is written

Three promises, kept everywhere. Example-first: every idea is introduced by something you can compute before it is stated in general. Two-sided examples: every statement comes with an example where it works and one where its hypotheses fail — the second kind is the course. No unexplained notation: every symbol is defined the moment it appears, because strong practical mathematicians should not need an Analysis course to read this.

"A probability of zero does not mean impossible. Learning what it does mean is why this course exists."

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