The Core Machinery: Measure and Integration

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.