Integration in a Probability Space
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There are two basic avenues to integration. In the modern approach the integral is introduced first for simple functions—as a weighted average of the values of the function-and then defined for any nonnegative measurable function f as a limit of the integrals of simple nonnegative functions increasing to f Conceptually this is extremely simple, but a certain price is paid in terms of proofs. The alternative classical approach, while employing a less intuitive definition, achieves considerable simplicity in proofs of elementary properties.
KeywordsRandom Walk Probability Space Simple Random Walk Monotone Convergence Theorem Markov Inequality
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