Problem set 8, submission of solutions NOT required

Prof. N. Kajino, Probability Theory WS 2012/2013

Problem set 8, submission of solutions NOT required

TheProblems below will be discussed in the tutorial on06.11.2012.

Problem 8.1. Let.X;M/be a measurable space and letf; g W X ! Œ 1;1 be M-measurable. Prove that the following sets belong toM:

¹x2X jf .x/ < g.x/º; ¹x2X jf .x/Dg.x/º; ¹x2X jf .x/ > g.x/º: Problem 8.2. Find the limits asN ! 1of the following series:

(1) X1 nD1

2 ⁿ

1C sin.2^Nn/

N 1

(2) X1 nD1

1

n.nCN / (3) X1 nD1

1C n

N N

Problem 8.3. Find the limits asn! 1of the following integrals:

(1) Z ₁

0

1

1Cxⁿdx (2) Z ₁

0

sine^x

1Cnx²dx (3) Z 1

0

ncosx 1Cn²x³⁼²dx Problem 8.4. Let.X;M; /be a measure space.

(1) Let fn W X ! Œ 1;1 be M-measurable for each n 2 Nand suppose that P₁

nD1

R

Xjfnjd <1. Prove that lim_n!1fn.x/D0for-a.e.x2X.

(2) ([1, Section 4.3, Problem 1]) Letf 2L¹./and¹fnº¹nD1L¹./. Suppose that fn 0onX for anyn 2 N, that lim_n!1fn.x/ D f .x/for anyx 2 X, and that lim_n!1R

XfndDR

Xf d. Prove that limn!1

R

Xjf fnjdD0.

Problem 8.5. Let.X;M/be a measurable space.

(1) Let S be a set, letA 2^S and let f W X ! S. Prove that f isM=S.A/- measurable if and only iff ¹.A/2Mfor anyA2A.

(2) Letn 2 N, and for eachi 2 ¹1; : : : ; nºlet .Si;Bi/be a measurable space and let fi W X ! Si. Prove that the map f WD .f1; : : : ; fn/ W X ! S1 Sn

isM=B1˝ ˝Bn-measurable if and only iffi isM=Bi-measurable for anyi 2

¹1; : : : ; nº.

(3) Let d 2 N and let f D .f1; : : : ; fd/ W X ! R^d, where fi W X ! R for each i 2 ¹1; : : : ; dº. Prove thatf isM=B.R^d/-measurable if and only if fi isM- measurable for anyi2 ¹1; : : : ; dº. (Proposition 2.24-(2) may be used.)

Problem 8.6. Let.X;M; /; .Y;N; /be-finite measure spaces, letf WX !Rbe M-measurable and letgWY !RbeN-measurable. Definef ˝gWXY !Rby .f ˝g/.x; y/WDf .x/g.y/. Prove the following statements:

(1)f ˝gisM˝N-measurable.

(2) Iff is-integrable andgis-integrable, thenf ˝gis-integrable and Z

XY

f ˝gd./D Z

X

f d Z

Y

gd:

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