- Let Ω = ℕ ={1,2,}. Let ℱ = P(ℕ), the set of all subsets of ℕ, and let μ(S) = number of elements in S. Thus μ({1}) = 1 = μ({2}),μ({1,2}) = 2, etc. In this case, all functions are measurable. For a nonnegative function, f defined on ℕ, show
What do the monotone convergence and dominated convergence theorems say about this example?

- For the measure space of Problem 1, give an example of a sequence of nonnegative measurable
functions {f
_{n}} converging pointwise to a function f, such that inequality is obtained in Fatou’s lemma. - If is a measure space and f,g ≥ 0 is measurable, show that if g= fa.e. ω, then ∫ gdμ = ∫ fdμ. Show that if f,g ∈ L
^{1}and g= fa.e. then ∫ gdμ = ∫ fdμ. - Let ,f be measurable functions with values in ℂ.converges in measure if
for each fixed ε > 0. Prove the theorem of F. Riesz. If f

_{n}converges to f in measure, then there exists a subsequence {f_{nk}} which converges to f a.e. In case μ is a probability measure, this is called convergence in probability. It does not imply pointwise convergence but does imply that there is a subsequence which converges pointwise off a set of measure zero. Hint: Choose n_{1}such thatChoose n

_{2}> n_{1}such thatn

_{3}> n_{2}such thatetc. Now consider what it means for f

_{nk}(x) to fail to converge to f(x). Use the Borel Cantelli lemma of Problem 14 on Page 443. - Suppose is a finite measure space ( μ< ∞) and S ⊆ L
^{1}. Then S is said to be uniformly integrable if for every ε > 0 there exists δ > 0 such that if E is a measurable set satisfying μ< δ, thenfor all f ∈S. Show S is uniformly integrable and bounded in L

^{1}if there exists an increasing function h which satisfiesS is bounded if there is some number, M such that

for all f ∈S. This is in the chapter but write it down in your own words.

- A collection S ⊆ L
^{1},a finite measure space, is called equiintegrable if for every ε > 0 there exists λ > 0 such thatfor all f ∈S. Show that S is equiintegrable, if and only if it is uniformly integrable and bounded. The equiintegrable condition is pretty popular in probability.

- Product measure is described in the chapter. Go through the construction in detail for two measure
spaces as follows.
Let K be the π system of measurable rectangles A × B where A ∈ℱ and B ∈G. Explain why this is really a π system. Now let ℱ×G denote the smallest σ algebra which contains K. Let

where both integrals make sense and are equal. Then show that P is closed with respect to complements and countable disjoint unions. By Dynkin’s lemma, P = ℱ×G. Then define a measure μ × ν as follows. For A ∈ℱ×G

Explain why this is a measure and why if f is ℱ×G measurable and nonnegative, then

Hint: Pay special attention to the way the monotone convergence theorem is used.

- Let be a regular measure space. For example, it could be ℝ with Lebesgue measure. Why do we care about a measure space being regular? This problem will show why. Suppose that closures of balls are compact as in the case of ℝ.
- Let μ< ∞. By regularity, there exists K ⊆ E ⊆ V where K is compact and V is open such that μ< ε. Show there exists W open such that K ⊆ ⊆ V and is compact. Now show there exists a function h such that h has values in,h= 1 for x ∈ K, and hequals 0 off W. Hint: You might consider Problem 12 on Page 442.
- Show that
- Next suppose s = ∑
_{i=1}^{n}c_{i}X_{Ei}is a nonnegative simple function where each μ< ∞. Show there exists a continuous nonnegative function h which equals zero off some compact set such that - Now suppose f ≥ 0 and f ∈ L
^{1}. Show that there exists h ≥ 0 which is continuous and equals zero off a compact set such that - If f ∈ L
^{1}with complex values, show the conclusion in the above part of this problem is the same.

- Let μ
- Let (Ω,ℱ,μ) be a measure space and suppose f,g : Ω → (−∞,∞] are measurable. Prove the
sets
are measurable. Hint: The easy way to do this is to write

Note that l

= x − y is not continuous on (−∞,∞] so the obvious idea doesn’t work. Heresignifies. - Let {f
_{n}} be a sequence of real or complex valued measurable functions. LetShow S is measurable. Hint: You might try to exhibit the set where f

_{n}converges in terms of countable unions and intersections using the definition of a Cauchy sequence. - Suppose u
_{n}(t) is a differentiable function for t ∈ (a,b) and suppose that for t ∈ (a,b),where ∑

_{n=1}^{∞}K_{n}< ∞. ShowHint: This is an exercise in the use of the dominated convergence theorem and the mean value theorem.

- Suppose is a sequence of nonnegative measurable functions defined on a measure space,. Show that
Hint: Use the monotone convergence theorem along with the fact the integral is linear.

- Show lim
_{n→∞}∑_{k=1}^{n}= 2 . This problem was shown to me by Shane Tang, a former student. It is a nice exercise in dominated convergence theorem if you massage it a little. Hint: - Show that if f is continuous on . Show that the traditional Riemann integral equals the Lebesgue integral. Hint: Assume first that f ≥ 0. You can reduce to this case later by looking at positive and negative parts. Show that there is an increasing sequence of step functions f
_{n}which converges uniformly to f on. Then from single variable calculus, or Theorem 4.2.1, and the monotone convergence theorem,Note how this uses the obvious fact that the Riemann integral and Lebesgue integral coincide on nonnegative step function since the Lebesgue meaure of an interval is just the length of the interval.

- Show the Vitali Convergence theorem implies the Dominated Convergence theorem for finite measure spaces but there exist examples where the Vitali convergence theorem works and the dominated convergence theorem does not.

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