What Is Randomness Mathematically?
Central Question
How does a mathematical model represent uncertainty without confusing outcomes, variables and distributions?
Why This Matters
Probability spaces separate the underlying experiment from observable quantities and provide a common foundation for discrete and continuous randomness.
Learning Prompt
I know undergraduate probability and real analysis. Reconstruct a probability space by asking what outcomes, observable events and consistent probabilities should mean. Motivate the sigma-algebra and probability measure rather than beginning with a definition. Develop random variables as measurable functions and distributions as the probabilities they induce, using a finite experiment and a continuous example. Make me distinguish an outcome, a random variable and its law, and explain why a distribution need not have a density. Keep the emphasis on what information the model captures and what assumptions it makes.