Guide · Track 05
Probability and measure
Measure theory and probability from the ground up: measures, σ-algebras, Lebesgue measure and the Lebesgue integral, convergence theorems, then probability spaces, distributions, expectation, independence, limit theorems and conditioning, and finally how they meet in option pricing.
Your progress0 of 13 read
13 lessons, in order
Each builds on the one before
- Measuring the size of setsWhat counting, length, area and mass have in common: the rules of a measure, why additivity must be countable, and the first examples of measures.Lebesgue MeasureUp next
- σ-algebras and Borel setsWhich sets receive a size: σ-algebras as collections of answerable questions, partitions and atoms, generated σ-algebras, Borel sets and Vitali's set.σ-Algebras
- Lebesgue measure and null setsBuilding length for every Borel set: countable covers and outer measure, sets of length zero, the Cantor sets, almost everywhere, and why length is unique.Lebesgue Measure
- Measurable functionsFunctions whose level sets can be measured: preimages, the half-line test, closure under limits, simple functions, σ(f) and the image measure.Random Variables
- The Lebesgue integralIntegrals built from simple functions by slicing the range: the layer-cake formula, integrable functions, sums as integrals, and Riemann versus Lebesgue.Lebesgue Integral
- Convergence theoremsWhen the limit of integrals is the integral of the limit: escaping mass, monotone convergence, Fatou's lemma, dominated convergence and series of integrals.Convergence
- Probability spaces and Kolmogorov's axiomsProbability as a measure of total mass one: outcomes, events and axioms, discrete and continuous models, infinite coin tossing, almost sure events.Probability Space
- Random variables and their distributionsRandom variables as measurable functions and the distributions they carry: distribution functions, atoms and densities, common laws, quantiles, sampling.Random Variables
- Expectation, variance and inequalitiesExpectation as an integral, computed from the distribution or from tails; linearity, variance and moments; the Markov, Chebyshev and Jensen inequalities.Lebesgue Integral
- Independence and product measuresIndependent events, σ-algebras and random variables, covariance and the variance of sums, product measures, Fubini's theorem and independent coin tosses.σ-Algebras
- Modes of convergence and the law of large numbersAlmost sure, in probability, in mean and in distribution: how they differ, the Borel–Cantelli lemma, uniform integrability and the laws of large numbers.Convergence
- Conditional probability and conditional expectationConditioning on an event, Bayes' rule, block averages over a partition, conditional expectation given a σ-algebra, best prediction and the tower property.Conditional Expectation
- Measure theory in option pricingHow the measure theory and probability of this track meet in option pricing: equivalent measures and densities, information, conditional expectation, limits.Probability Space · σ-Algebras · Conditional Expectation