MAST90012 Measure Theory Semester 1, 2015

Assignment 2

Due: Friday May 29 th

All answers should be appropriately justified.

1. Let (X,A,µ) and (Y,B,?) be s-finite measure spaces. Let E ? A?B be such that µ(Ey) = 0 ?-almost everywhere. Show that ?(Ex) = 0 µ-almost everywhere.

2. Let (X,A,µ) and (Y,B,?) be s-finite measure spaces. Let f ? L1(X,A,µ) and g ? L1(Y,B,?). For x ? X and y ? Y define ?(x,y) = f(x)g(y). Show that ? ? L1(X × Y,A ? B,µ × ?) and that

3. Show that

for s 0 by integrating e-sx sin2xy with respect to x and y.

4. (a) Let A = {(x,0) | -1 6 x 6 1}. Show, from the definition of Hausdorff measure, that H2(A) = 0.

(b) The Sierpinski carpet is defined in a manner analogous to the Sierpinski triangle, but beginning with a square. The set is given by C = ni 0Ci where the Ci ? Ci+1 and C0,C1 and C2 are indicated below.

C0 C1 C2

(i) Show that the Sierpinski carpet has Lebesgue measure 0.

(ii) Show that the Sierpinski carpet has Hausdorff dimension log3 8.

5. Let X be a Polish space and A ? BX and uncountable Borel subset of X. Show that A contains a subset that is homeomorphic to Cantor space. (We had this statement in lectures with an outline of the proof. This question is asking you in fill in the details.)

6. Let µ be a Radon measure on an LCH space X.

(a) Let N be the union of all open U ? X such that µ(U) = 0. Show that µ(N) = 0. The complement of N is called the support of µ, supp(µ).

(b) Show that x ? supp(µ) if and only if R fdµ 0 for every f ? Cc(X,[0,1]) such that f(x) 0.

Assignment 2

Due: Friday May 29 th

All answers should be appropriately justified.

1. Let (X,A,µ) and (Y,B,?) be s-finite measure spaces. Let E ? A?B be such that µ(Ey) = 0 ?-almost everywhere. Show that ?(Ex) = 0 µ-almost everywhere.

2. Let (X,A,µ) and (Y,B,?) be s-finite measure spaces. Let f ? L1(X,A,µ) and g ? L1(Y,B,?). For x ? X and y ? Y define ?(x,y) = f(x)g(y). Show that ? ? L1(X × Y,A ? B,µ × ?) and that

3. Show that

for s 0 by integrating e-sx sin2xy with respect to x and y.

4. (a) Let A = {(x,0) | -1 6 x 6 1}. Show, from the definition of Hausdorff measure, that H2(A) = 0.

(b) The Sierpinski carpet is defined in a manner analogous to the Sierpinski triangle, but beginning with a square. The set is given by C = ni 0Ci where the Ci ? Ci+1 and C0,C1 and C2 are indicated below.

C0 C1 C2

(i) Show that the Sierpinski carpet has Lebesgue measure 0.

(ii) Show that the Sierpinski carpet has Hausdorff dimension log3 8.

5. Let X be a Polish space and A ? BX and uncountable Borel subset of X. Show that A contains a subset that is homeomorphic to Cantor space. (We had this statement in lectures with an outline of the proof. This question is asking you in fill in the details.)

6. Let µ be a Radon measure on an LCH space X.

(a) Let N be the union of all open U ? X such that µ(U) = 0. Show that µ(N) = 0. The complement of N is called the support of µ, supp(µ).

(b) Show that x ? supp(µ) if and only if R fdµ 0 for every f ? Cc(X,[0,1]) such that f(x) 0.

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