Finite Probability Spaces

Definition 1.10 (measure, Lebesgue measure) LetExampleDefinition 1.11 (Borel-measurable)We say rv. X is Borel-measurable if σ ( X ) is a subset of Borel set.Definition 1.12 (Lebesgue integral)A measure that equal to the length of the interval.The Lebesgue integral has two important advantages over the Riemann integral.The first is that the Lebesgue integral is defined for more functions.The second is that there are three convergence theorems satisfied by the Lebesgueintegral. But there are no such theorems for the Riemann integral,ExamplePlease see ShreveTheorem 3.1 (Fatou’s Lemma)Theorem 3.2 (Monotone Convergence Theorem)Theorem 3.3 (Dominated Convergence Theorem)