Chapter 3 - The Integral

Question 1

Define a simple function.

Define the preintegral of a simple function.

Define the integral of a positive measurable function.

Answer 1

Question 2

State Proposition 3.1 on properties of the preintegral.

State Lemma 3.1 on integral vs preintegral of simple functions.

State Lemma 3.2 - New Measures from Old.

Answer 2

Question 3

State Lemma 3.3 - Monotonicity of the Integral.

State THM 3.1 - Monotone Convergence Theorem.

Answer 3

Question 4

State Lemma 3.4 - What does the MCT mean for our choice of simle functions phi_n?

State Corollary 3.1 on linearity of the integral.

Answer 4

Question 5

The MCT only holds for increasing functions. State THM 3.2 - Fatou’s Lemma on weaker assumptions and conclusions.

What does it mean for a statement to be satisfied for ‘mu-almost everywhere’/’mu-almost surely’?

Answer 5

Question 6

State Corollary 3.2 on a function being 0 mu-a.e.

State Corollary 3.3 - the a.s. verison of MCT.

Answer 6

Question 7

State Corollary 3.4 on integrals of sums/ sums of integrals.

State Corollary 3.5 on ‘More New Measures from Old.’

Answer 7