Probability and Measure

Probability and Measure(D)

PROBABILITY AND MEASURE (D)
Analysis II is essential.

  • Measure spaces, σ-algebras, π-systems and uniqueness of extension, statement ∗and proof∗ of Carath´eodory’s extension theorem. Construction of Lebesgue measure on R. The Borel σ-algebra of R. Existence of non-measurable subsets of R. Lebesgue-Stieltjes measures and probability distribution functions. Independence of events, independence of σ-algebras. The Borel–Cantelli lemmas. Kolmogorov’s zero-one law. [6]
  • Measurable functions, random variables, independence of random variables. Construction of the integral, expectation. Convergence in measure and convergence almost everywhere. Fatou’s lemma, monotone and dominated convergence, differentiation under the integral sign. Discussion of product measure and statement of Fubini’s theorem. [6]
  • Chebyshev’s inequality, tail estimates. Jensen’s inequality. Completeness of Lp for 1 6 p 6 ∞. The H¨older and Minkowski inequalities, uniform integrability. [4]
  • L2 as a Hilbert space. Orthogonal projection, relation with elementary conditional probability. Variance and covariance. Gaussian random variables, the multivariate normal distribution. [2]
  • The strong law of large numbers, proof for independent random variables with bounded fourth moments. Measure preserving transformations, Bernoulli shifts. Statements ∗and proofs∗ of maximal ergodic theorem and Birkhoff’s almost everywhere ergodic theorem, proof of the strong law. [4]
  • The Fourier transform of a finite measure, characteristic functions, uniqueness and inversion. Weak convergence, statement of L´evy’s convergence theorem for characteristic functions. The central limit theorem. [2]

Appropriate books

  • P. Billingsley Probability and Measure. Wiley 1995
  • R.M. Dudley Real Analysis and Probability. Cambridge University Press 2002
  • R.T. Durrett Probability: Theory and Examples. Wadsworth and Brooks/Cole 1991
  • D. Williams Probability with Martingales. Cambridge University Press

Associated GitHub Page
https://jaircambridge.github.io/Probability-and-Measure-D/