Integration: Exercises

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 8 March 2011

Measure Theory Problem Set 2

  1. Let (X,Σ) be a measure space. Let f,g:X be measurable. Let ε>0. Show that the set {xX | f(x)+ε< g(x) } is in Σ.
  2. Let (X,Σ) be a measure space and let μ:Σ[0,] be a measure.
    (i)   For h:X a simple function, give the definition of hdμ.
    (ii)   Let f:X 0 be integrable. Give the definition of hdμ.
    (iii)   Working directly from the definition of the integral show that if g:X is defined by g(x)= 3f(x) then gdμ =3 fdμ .
  3. Let K be a compact subset of . Let f:K be continuous. Let ε>0. Show that there exist n0 and a0, a1,, an such that for all xK, | f(x)-( a0+ a1x+ a2x2+ + anxn) | <ε.
  4. Show that if μ,ν are Borel probability measures on a standard Borel space then we can find (positive) μ0 ,μ1 with μ=μ0 +μ1 μ0ν, μ1ν.
  5. Let m* be the outer measure used to define Lebesgue measure m. Carefully define m* and Lebesgue measurable sets. Show that if B is Lebesgue measurable then there are Borel A1,A2 with A1 BA2 and m(A2 \A1) =0.
  6. Let Σ0 Σ1 be two σ-algebras on a set X. Let μ be a σ-finite measure on (X,Σ1) and let f:X be measurable with respect to Σ1. Show that there is a function g:X which is measurable with respect to Σ0 such that on any BΣ0 B g(x) dμ(x) = B f(x) dμ(x)
  7. Let Y be a standard Borel space and let μ be a finite atomless Borel measure on Y. Then for all c with 0cμ(Y), we can find BY with μ(B) =c.
  8. Show that there exists a subset V[0,1] which is not Lebesgue measurable.
  9. Let K be a compact metric space. Let AK be Borel. Let ν1, ν2 be finite Borel measures on K with the property that ν1(B)=0 for all Borel BAc, and ν2(B)=0 for all Borel BA. Let μ be the signed measure ν1-ν2 (i.e. μ(B)= ν1(B) -ν2(B)). Show that ν1(A) +ν2( Ac) equals the supremum of {|fdμ| | f:K[-1,1] is continuous}.
  10. Let K =[0,1], the unit interval in its usual topology. Let P(K) be the space of probability measures on K equipped with the topology generated by the basic open sets { μP(K) | f1dμ (a1,b1) , f2dμ (a2,b2) , , fndμ (an,bn) }, for n>0, f1,, fn continuous functions from K to , and a1,b1, , an,bn . Let U be the open interval (0, 12). Is A={μP(K) | μ(U)=1} closed in this topology?
  11. Let (X,Σ) be a finite measure space. Let (fn) n>0 be a sequence of measurable functions which converge pointwise to f. Then for any ε>0 there is AΣ with μ(X\A)<ε and (fn) converging uniformly to f on A.
  12. Equip {H,T} with the discrete topology and let >0 {H,T} be the collection of all functions f: >0 {H,T} equipped with the product topology. For i= (i1, i2, iN) a sequence of distinct elements of >0 and S= (S1, S2,, SN) a sequence of elements of {H,T} let A i, S ={ f >0 | f(i1)=S1, f(i2)=S2, , f(iN)=SN }. Let Σ0 be the collection of all finite unions of the sets of the form A i, S . Define μ0 : { A i, S } 0 by μ0 ( A i, S )= 2-N. Show that μ0 extends to a function on Σ0 which is σ-additive on its domain.

Notes and References

These exercises are taken from problems by G. Hjorth for a course in "Measure Theory" at the Masters level at University of Melbourne.


[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR??????.

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