Measure Theory // 1

Explain the paradoxes that arise when trying to assign equal probability to all points in a continuous interval, leading to the idea of non-measurable sets.

Describe how a 'measure' generalizes the concepts of length, area, and volume to more abstract sets.

Identify the limitations of Riemann integration that motivate the need for a more powerful theory of integration for probability.

Define a σ-algebra and explain its role in specifying the collection of 'valid' events in a probability model.

Define a probability space as the triple (Ω, F, P) and explain the role of each component: sample space, σ-algebra of events, and probability measure.

Describe a random variable formally as a 'measurable function' and explain what this means intuitively in terms of preserving structure.

Contrast the Lebesgue integral with the Riemann integral conceptually, focusing on the difference between partitioning the domain versus the codomain.

Explain the conceptual importance of the Dominated Convergence Theorem for justifying the interchange of limits and expectations in statistical theory.

Explain how the Strong Law of Large Numbers relies on measure theory for its precise statement and proof.

Define conditional expectation with respect to a σ-algebra, and interpret the σ-algebra as the 'information' available.

Define a filtration as an increasing sequence of σ-algebras that formally models the flow of information over time.

Define a martingale with respect to a filtration and explain its connection to the concepts of a 'fair game' and efficient markets.

Relate the measure-theoretic concepts of filtrations and martingales to the analysis of stochastic processes from your background in finance and time series.