Fundamentals of Measure Theory, Problem Set 2
1. Let µ denote Lebesgue measure on R.
(a) Let {an }n∈N be an enumeration of Q ∩ [0, 1]. For any ε > 0, use this enumeration
to give an explicit formula for a closed subset F ⊂ [0, 1]rQ such that µ(F ) > 1−ε.
(b) Is there a closed subset of [0, 1] r Q with Lebesgue mesaure 1?
2. Let µ denote Lebesgue measure on R.
(a) Show that for all 0 < α < 1 there is a closed subset E of [0, 1] such that µ(E) = α
and E does not contain any interval. Do so by constructing E as a Cantor-like set,
only that the middle part which is removed at each stage would be smaller than
1/3 of the interval.
(b) Show that there exists a measurable subset E of [0, 1] such that for each subinterval
J of [0, 1] we have 0 < µ(J ∩ E) < µ(J). Hint: construct an infinite union of
Cantor-like subsets of various intervals.
3. Suppose that E, F are Borel subsets of R with positive Lebesgue measure. Show that
E + F contains an interval.
4. Let f : R → R an additive function, that is, f (x + y) = f (x) + f (y) for all x, y.
(a) Show that f is Q-linear, i.e. f (rx) = rf (x) for any rational number r.
(b) Show that if f is continuous at 0 then f (x) = Ax for all x, where A = f (1).
(c) Show that if there is an interval (−a, a) for a > 0 in which f is bounded then f is
continuous at 0.
(d) Show that if f is measurable then f (x) = Ax (hint: define Em = {x | |f (x)| < m}.
Show that Em has positive measure for some m, and consider the set Em − Em .)
5. Show that if f : R → R is monotone then f is Borel measurable.
6. Let f : R → R be a function. Show that the set of all points at which f is continuous
is an intersection of countably many open sets (and therefore this set is in the Borel
σ-algebra).
Remark: Sets which are an intersection of countably many open sets are called Gδ sets.
7. Let (Ω, B) be a measurable space. Let fn : Ω → R be a sequence of measurable functions.
Show that {x | lim fn (x) exists} is a measurable set.
n→∞
8. Suppose (Ω1 , B1 ), (Ω2 , B2 ) are measurable spaces.
(a) Suppose F : Ω1 → Ω2 is measurable. Let µ be a measure on (Ω1 , B1 ). For
each E ∈ B2 , define ν(E) = µ(F −1 (E)). Show that ν is a measure on (Ω2 , B2 ).
The measure ν is called the push-forward measure of µ with respect to F , and
is sometimes denoted F∗ µ. (Remark: if (Ω1 , B1 , µ) is a probability space, and
F : Ω1 → R) then in probability theory F is called a random variable, and the
push-forward measure is called the probability distribution of F .)
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(b) Let Ω = {0, 1}n with normalized counting measure - P
the measure of each point is
n
1/2n . Define F : Ω → R by F ((ω1 , ω2 , ..., ωn )) = n1 i=1 ωi . Describe the pushforward measure of F .
Remark: notice that this space can be seen as the sample space for all possible
outcomes of n coin tosses of a fair coin, where 1 indicates “heads” and 0 indicates
“tails”, and F counts the percentage of “heads” in a given outcome.
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