MTH309A: Probability Theory

$\sigma$-field of events (emphasis on $\sigma$-fields generated by a collection of sets, Borel $\sigma$-fields), Limits of sequences of sets. Principle of good sets (Monotone Class Theorem, Dynkin's $\pi-\lambda$ Theorem).

Measure and Measure space (emphasis on probability measure and probability space), properties, Borel-Cantelli Lemma (first half).

Extension of measures (existence and uniqueness), Outer Measure, Carath\'eodery Extension Theorem for $\sigma$-finite measure. Completeness of measure spaces.

Measurable functions (emphasis on Random variables), algebraic properties and limits of measurable functions. Probability distribution of Random variables, induced probability space, distribution function.

Correspondence between distribution functions and Lebesgue-Stieltjes measures (emphasis on probability measures), construction of Lebesgue measure.

Integration of measurable functions (emphasis on Expectation and moments of Random variables, change of variables/measures), inequalities.

Convergence theorems for expectations of sequences of random variables (monotone convergence theorem, Fatou's lemma, dominated convergence theorem).

Connection between Riemann and Lebesgue Integration.

Various modes of convergence of sequences of random variables (almost surely, in probability, in $r$-th mean). $L^p$ spaces.

Independence of events and random variables. Borel-Cantelli Lemma (second half).

Convergence of series of independent random variables, Kolmogorov inequality, Kolmogorov three-series
criterion.

Characteristic function and its properties, inversion formulae.

Weak convergence of probability measures, connection with convergence of sequences of distribution functions, Helly-Bray theorems, Portmanteu Theorem.

Khintchin's weak law of large numbers, Kolmogorov strong law of large numbers. Central limit
theorems of Lindeberg-Levy, Liapounov and Lindeberg-Feller.