Math 540-Real Analysis- Homework 2
Due date: September 17-Submission in pairs
(1) Let µbe an additive measure on an algebra R. Assume in addition that µ
is σfinite. Let Σµbe the algebra of measurable sets (constructed from the
outer measure µ∗). Show that for every B∈Σµthere exists A∈ Rσ,δ such
that B⊂Aand µ∗(A\B) = 0. Is the assumption µ σ-finite necessary?
(2) Show that for every set B⊂Rwith m∗(B)>0 there exists a non-
measurable set E⊂B.
(3) Let Fbe a monotone increasing function, F(−∞) = 0 and Fis right
continuous, i.e. limh→0,h>0F(a+h) = F(a). Let a1< a2be such that
F(a2) = F(a1). Show that every set A⊂(a1, a2) we have µ∗
F(A) = 0.
What about the endpoints?
(4) Let Rbe a σ-algebra and µbe an additive measure which is σ-additive on
R. Let µ1and µ2be two measures on B(R) such that µ1|R =µ1|R. Show
that µ1=µ1. How can extension on Σµdiffer?
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