- 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 ≥ 0 is measurable, show that if g= fa.e. ω and g ≥ 0, 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 1457. - 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.

- 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.

- There is a general construction called product measure. You have two finite measure
spaces.
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: This is just a repeat of what I showed you in class except that it is easier because the measures are finite. Pay special attention to the way the monotone convergence theorem is used.

- Let be a regular measure space. For example, it could be ℝ
^{p}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^{p}.- 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 1457.
- 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.

- Explain why for each t > 0,x → e
^{−tx}is a function in L^{1}andThus

Now explain why you can change the order of integration in the above iterated integral. Then compute what you get. Next pass to a limit as R →∞ and show

This is a very important integral. Note that the thing on the left is an improper integral. sin

∕t is not Lebesgue integrable because it is not absolutely integrable. That isIt is important to understand that the Lebesgue theory of integration only applies to nonnegative functions and those which are absolutely integrable.

- 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: - Let the rational numbers in be
_{k=1}^{∞}and defineShow that lim

_{n→∞}f_{n}= fwhere f is one on the rational numbers and 0 on the irrational numbers. Explain why each f_{n}is Riemann integrable but f is not. However, each f_{n}is actually a simple function and its Lebesgue and Riemann integral is equal to 0. Apply the monotone convergence theorem to conclude that f is Lebesgue integrable and in fact, ∫ fdm = 0. - Give an example of a sequence of functions ,f
_{n}≥ 0 and a function f ≥ 0 such that f= liminf_{n→∞}f_{n}but ∫ fdm < liminf_{n→∞}∫ f_{n}dm so you get strict inequality in Fatou’s lemma. - Let f be a nonnegative Riemann integrable function defined on . Thus there is a unique number between all the upper sums and lower sums. First explain why, if a
_{i}≥ 0,Explain why there exists an increasing sequence of Borel measurable functions

converging to a Borel measurable function g, and a decreasing sequence of functionswhich are also Borel measurable converging to a Borel measurable function h such that g_{n}≤ f ≤ h_{n},dm = 0. Explain whyis a set of measure zero. Then explain why f is measurable and ∫_{a}^{b}fdx = ∫ fdm so that the Riemann integral gives the same answer as the Lebesgue integral.

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