Measure Theory and Metric Spaces

Construct the sigma-algebra generated by a collection of sets in finite and countable examples.

Identify the Borel sigma-algebra generated by the open sets of a metric space.

Test measurability of a function using inverse images of generating sets.

Prove that a continuous map between metric spaces is Borel measurable.

Determine measurability of functions formed by algebraic operations, suprema, and pointwise limits.

Distinguish continuity from Borel measurability using concrete examples.

Verify whether a set function is a measure on a specified sigma-algebra.

Compute measures of limiting set sequences using continuity from below or continuity from above after checking its finiteness condition.

Compute outer measures of simple subsets of the real line using countable coverings.

Apply the Carathéodory criterion to test measurability in elementary examples.

Trace how Lebesgue measure is obtained by restricting Lebesgue outer measure.

Construct the completion of a measure and distinguish Borel measure from completed Lebesgue measure using null sets.

Compute the Lebesgue integral of a nonnegative simple function.

Construct increasing simple-function approximations to a nonnegative measurable function.

Apply the Monotone Convergence Theorem to evaluate a limit of integrals.

Use Fatou’s lemma to obtain a valid bound when integral convergence is uncertain.

Determine whether a signed measurable function is integrable and compute its integral from its positive and negative parts.

Apply the Dominated Convergence Theorem after verifying each of its hypotheses.

Justify an interchange of an infinite sum and an integral using an appropriate convergence theorem.

Interpret equality almost everywhere and identify functions that represent the same element of an Lp space.

Compute Lp norms for concrete functions and random-variable examples for 1 ≤ p ≤ ∞.

Apply Hölder’s inequality to bound an integral.

Apply Minkowski’s inequality to bound an Lp norm.

Distinguish absolute continuity from singularity for finite measures in concrete examples.

Apply the Radon–Nikodym theorem to identify and compute derivatives in elementary discrete and continuous settings.

Construct the product sigma-algebra from measurable rectangles.

Compute a product measure on rectangles and finite unions of rectangles.

Apply Tonelli’s theorem to a nonnegative measurable function.

Apply Fubini’s theorem to an integrable function.

Diagnose why an interchange of integration fails when Fubini’s hypotheses are absent.

Model a concrete random experiment as a probability space.

Verify that a proposed random variable is measurable.

Compute the distribution of a random variable as a pushforward measure.

Relate probability mass functions, densities, and cumulative distribution functions to their underlying probability measures.

Compute expectations as Lebesgue integrals in discrete, continuous, and mixed examples.

Use the law of the unconscious statistician to evaluate an expectation after a transformation.

Compute moments or tail probabilities from a distribution and determine when they are finite.

Apply Markov’s and Chebyshev’s inequalities to obtain probability bounds from moment information.

Recover marginal distributions from a joint distribution.

Verify independence using product measures, joint densities, or generating events.

Distinguish almost-sure, in-probability, Lp, and in-distribution convergence in concrete sequences.

Establish standard implications among modes of convergence and identify counterexamples to their converses.

Apply the Borel–Cantelli lemmas to determine whether events occur infinitely often.

Reconstruct the finite-variance weak law of large numbers proof using Chebyshev’s inequality.

Apply the strong law of large numbers to iid sample averages and contrast its conclusion with the weak law.

Use the central limit theorem to approximate sampling probabilities and the distribution of a standardized sample mean.

Verify a candidate conditional expectation using its defining integral identity and identify its almost-sure uniqueness.

Compute conditional expectation with respect to a finite measurable partition.

Explain how the Radon–Nikodym theorem yields the existence of conditional expectation.

Compute conditional expectations from a discrete joint distribution.

Compute conditional expectations from a joint density.

Apply the pull-out property when a factor is measurable with respect to the conditioning sigma-algebra.

Apply the tower property to simplify iterated conditional expectations.

Interpret conditional expectation in L2 as an orthogonal projection onto available information.

Describe a discrete-time stochastic process through its sample paths and finite-dimensional distributions.

Determine whether a process is adapted to a given filtration.

Verify whether a discrete random time is a stopping time for a filtration.

Verify the martingale, submartingale, or supermartingale property from conditional expectations.

Construct martingales from random walks and conditional expectations of integrable random variables.

Compute conditional and unconditional expectations of martingale increments using the martingale-difference property.

Apply an optional stopping theorem after checking its stated hypotheses.

Use a counterexample to explain why optional stopping can fail without suitable hypotheses.