Kenneth Kuttler

11.16. EXERCISES 355

26. Find the area of the bounded region R, determined by 5x+ y = 1,5x+ y = 9,y = 2x,and y = 5x.

27. Here are three vectors. (1,2,3)T ,(1,0,1)T , and (2,1,0)T . These vectors determinea parallelepiped, R, which is occupied by a solid having density ρ = y. Find themass of this solid. To find the mass of the solid, you integrate the density. Thus, ifP is this parallelepiped, the mass is

∫P ydm3 Hint: Let h : [0,1]3 → P be given by

h(t1, t2, t3) = t1

 123

+ t2

 101

+ t3

 210

 then by definition of what is meant

by a parallelepiped, h([0,1]2

)= P and h is one to one and onto.

28. Suppose f ,g ∈ L1 (Rp) . Define f ∗ g(x) by∫

f (x−y)g(y)dmp (y) . First showusing the preceding problem that there is no loss of generality in assuming thatboth f ,g are Borel measurable. Next show this makes sense a.e. x and that infact for a.e. x,

∫| f (x−y)| |g(y)|dmp (y) < ∞. Next show

∫| f ∗g(x)|dmp (x) ≤∫

| f |dmp∫|g|dmp. Hint: You can use Fubini’s theorem to write∫ ∫

| f (x−y)| |g(y)|dmp (y)dmp (x) =∫ ∫| f (x−y)| |g(y)|dmp (x)dmp (y) =

∫| f (z)|dmp

∫|g(y)|dmp.

29. Suppose X : (Ω,F ,P) where P is a probability measure and suppose X : Ω→ R ismeasurable. That is, X−1 (open set)∈F . Then consider the distribution measure λ Xdefined on the Borel sets of Rp and given as follows. λ X (E) = P(X ∈ E). Explainwhy this is a probability measure on B (R) and why X−1 (B) ∈F whenever B is aBorel set. Next show that if X ∈ L1 (Ω) ,

∫Ω

XdP =∫R xdλ X . Suppose h is a complex

valued Borel measurable function defined on R which is also bounded. Show that∫h(x)dλ X (x) =

∫h(X (ω))dP

Hint: Recall that from the definition of the integral,∫R|x|dλ X =

∫∞

0λ X (|x|> α)dα =

∫∞

0P(|X |> α)dα =

∫Ω

|X |dP < ∞

30. Let h : U → h(U) be one to one and C1. Use the inverse function theorem to give amuch easier proof of the change of variables formula.

31. If a continuous function is one to one on a compact set, explain why its inverse iscontinuous.

32. Suppose U is a nonempty set in Rp. Let ∂U consist of the points p ∈ Rp such thatB(p,r) contains points of U as well as points of Rp \U . Show that U is contained inthe union of the interior of U, denoted as int(U) with ∂U . Now suppose that f : U→Rp and is one to one and continuous. Explain why int(f (U)) equals f (int(U)).

33. Prove the Radon Nikodym theorem, Theorem 11.13.2 in case λ ≪ µ another wayby using the earlier general Radon Nikodym theorem, Theorem 10.13.7 or its corol-lary and then identifying the function ensured by that theorem with the symmetricderivative, using the fundamental theorem of calculus, Theorem 11.4.2.

11.16. EXERCISES 35526.27.28.29.30.31.32.33.Find the area of the bounded region R, determined by 5x+y = 1,5x+y=9,y = 2x,and y = 5x.Here are three vectors. (1,2,3)’ ,(1,0,1)’, and (2,1,0)". These vectors determinea parallelepiped, R, which is occupied by a solid having density p = y. Find themass of this solid. To find the mass of the solid, you integrate the density. Thus, ifP is this parallelepiped, the mass is {pydm3 Hint: Let h : (0, 1}> — P be given by1 1 2h(t,f2,0)=t {| 2 J|+m 1 0 }+] 1 | then by definition of what is meant3 1 0by a parallelepiped, h ((0. I?) = P and his one to one and onto.Suppose f,g € L!(R?). Define f * g(x) by { f(x—y)g(y)dmp (y). First showusing the preceding problem that there is no loss of generality in assuming thatboth f,g are Borel measurable. Next show this makes sense a.e. 2 and that infact for ae. x, [| f(a—y)||g (y)|dmp(y) < oe. Next show f|f* 9 (ax)|dmp (a) <J \f|dmp f |g|dm,. Hint: You can use Fubini’s theorem to writeJ [if @ wile lamp (9) mp (x) =[ [\f@=whle lay (e)dmy(y) = [ |r (z)\dmy [lg (w)\amp.Suppose X : (Q,.F,P) where P is a probability measure and suppose X : QO — R ismeasurable. That is, X~' (open set) € ¥. Then consider the distribution measure Aydefined on the Borel sets of R? and given as follows. Ax (E) = P(X € E). Explainwhy this is a probability measure on A(R) and why X~! (B) € F whenever B is aBorel set. Next show that if X € L' (Q) , [5 XdP = JexdAy. Suppose h is a complexvalued Borel measurable function defined on R which is also bounded. Show that[res )day (x )= [nx(oHint: Recall that from the definition of the integral,| Ix|dax =| Ax (\x| >a)da= | P(|\X|> ada = | IX|dP <0R 0 0 QLet h : U -+ h(U) be one to one and C!. Use the inverse function theorem to give amuch easier proof of the change of variables formula.If a continuous function is one to one on a compact set, explain why its inverse iscontinuous.Suppose U is a nonempty set in R?. Let 0U consist of the points p € R? such thatB(p,r) contains points of U as well as points of R? \ U. Show that U is contained inthe union of the interior of U, denoted as int (U) with QU. Now suppose that f : U >R? and is one to one and continuous. Explain why int(f (U)) equals f (int(U)).Prove the Radon Nikodym theorem, Theorem 11.13.2 in case A < mW another wayby using the earlier general Radon Nikodym theorem, Theorem 10.13.7 or its corol-lary and then identifying the function ensured by that theorem with the symmetricderivative, using the fundamental theorem of calculus, Theorem 11.4.2.