Departments of Mathematics Fall 2015 Montana State University Prof. Kevin Wildrick Measure Theory Review Set 2 1. Define • a σ-algebra • a measure space • a measurable set in a measure space • a measurable function between two measure spaces • a measurable function from a measure space to the extended real line • a simple function • the standard representation of a simple function • the integral of a measurable simple function on a measure space • the integral of a non-negative measurable function on a measure space • the integral of a measurable function on a measure space 2. State and prove • the fundamental approximation lemma • the monotone convergence theorem • Fatou’s lemma • the “linearity” of the integral 3. Let (X, Σ, µ) be a measure space, and let {En }n∈N be a sequence of measurable subsets of X. Show that S P • µ( n∈N En ) ≤ n∈N µ(En ), • if E1 ⊆ E2 , then µ(E1 ) ≤ µ(E2 ). 4. Let f : R → R be a continuous function, and let [a, b] be a compact interval in R. For each integer N ≥ 1 and each i ∈ {1, . . . , N }, define (i − 1)(b − a) i(b − a) N Ii = a + ,a + N N and N aN . i = min f (x) : x ∈ Ii • Show that φN = N X i i=1 is a measurable simple function. aN i χI N • Show that Z Z lim N →∞ R b f (x) dx, φN dm = a where the right hand side above denotes the Riemann integral of f from a to b. • Show that {φN }N ≥1 is an increasing sequence and that lim φN = f. N →∞ • Show that Z b Z f dm. f (x) dx = [a,b] a 5. Let (X, Σ, µ) be a measure space, and suppose that E ∈ Σ satisfies µ(E) = 0. Show that for any measurable function f : (X, Σ, µ) → R, Z f dµ = 0. E (don’t forget to include a proof that the above integral is defined!) 6. Consider the measure space (N, P(N), µ), where every set is measurable and µ(E) = card E is the “counting measure”. Given a seqeuence {an }n∈N ⊆ [0, ∞], find a measurable function f : N → [0, ∞] such that Z X f dµ = an . N What happens if we allow {an }n∈N ⊆ R? n∈N