Calculus of Real and Complex Variables

5.6 Exercises

1. Show carefully that if S is a set whose elements are σ algebras which are subsets of P
   (Ω)
   , then ∩S is also a σ algebra. Now let G⊆P
   (Ω)
   satisfy property P if G is closed with respect to complements and countable disjoint unions as in Dynkin’s lemma, and contains ∅ and Ω. If ℌ is any set whose elements are G which satisfy property P, then ∩ℌ also satisfies property P.
2. Show ℬ
   (ℝn)
   = σ
   (P )
   where P consists of the half open rectangles which are of the form ∏ _i=1ⁿ[a_i,b_i).
3. Recall that f :
   (Ω,ℱ )
   → ℝ is measurable means f⁻¹
   (open)
   ∈ ℱ. Show that if E is any set in ℬ
   (ℝ)
   , then f⁻¹
   (E)
   ∈ ℱ. Thus, inverse images of Borel sets are measurable. Next consider f :
   (Ω, ℱ)
   → ℝ being measurable and g : ℝ → ℝ is Borel measurable, meaning that g⁻¹
   (open)
   ∈ℬ
   (ℝ)
   . Explain why g ∘ f is measurable. Hint: You know that
   (g ∘f)
   ⁻¹
   (U)
   = f⁻¹
   ( ) g− 1(U )
   .  For your information, it does not work the other way around. That is, measurable composed with Borel measurable is not necessarily measurable. In fact examples exist which show that if g is measurable and f is continuous, then g ∘ f may fail to be measurable. However, these things are not for this book.
4. Let X_i ≡ ℝ^n_i and let X = ∏_i=1ⁿX_i and let the distance between two points in X be given by

   ∥x − y∥ ≡ max {∥xi − yi∥,i = 1,2,⋅⋅⋅,n}

   Show that any set of the form

   n∏ Ei, Ei ∈ ℬ (Xi) i=1

   is a Borel set. That is, the product of Borel sets is Borel. Hint: You might consider the continuous functions π_i : ∏ _j=1ⁿX_j → X_i which are the projection maps. Thus π_i

   (x)
   ≡ x_i. Then π_i⁻¹
   (Ei)
   would have to be Borel measurable whenever E_i ∈ℬ
   (Xi )
   . Explain why. You know π_i is continuous. Why would π_i⁻¹
   (Borel)
   be a Borel set? Then you might argue that ∏ _i=1ⁿE_i = ∩_i=1ⁿπ_i⁻¹
   (Ei)
   .

5. You have two finite measures defined on ℬ
   (X )
   μ,ν. Suppose these are equal on every open set. Show that these must be equal on every Borel set. Hint: You should use Dynkin’s lemma to show this very easily.
6. Show that
   (ℕ,P (ℕ),μ)
   is a measure space where μ
   (S )
   equals the number of elements of S. You need to verify that if the sets E_i are disjoint, then μ
   (∪ ∞i=1Ei)
   = ∑ _i=1^∞μ
   (Ei)
   .
7. Let Ω be an uncountable set and let ℱ denote those subsets of Ω, F such that either F or F^C is countable. Show that this is a σ algebra. Next define the following measure. μ
   (A )
   = 1 if A is uncountable and μ
   (A)
   = 0 if A is countable. Show that μ is a measure.
8. Let μ
   (E)
   = 1 if 0 ∈ E and μ
   (E )
   = 0 if 0
   ∕∈
   E. Show this is a measure on P
   (ℝ)
   .
9. Give an example of a measure μ and a measure space and a decreasing sequence of measurable sets
   {Ei}
   such that lim_n→∞μ
   (En)
   ≠μ
   ∞ (∩ i=1Ei)
   .
10. If you have a finite measure μ on ℬ
    (ℝn )
    , and if F ∈ℬ
    (X )
    , show that there exist sets E,G such that G is a countable intersection of open sets and E is a countable union of closed sets such that E ⊆ F ⊆ G and μ
    (G ∖E )
    = 0.
11. You have a measure space
    (Ω, ℱ,P)
    where P is a probability measure on ℱ. Then you also have a measurable function X : Ω → ℝⁿ. Thus X⁻¹
    (U )
    ∈ℱ whenever U is open. Now define a measure on ℬ
    (ℝn)
    denoted by λ_X and defined by λ_X
    (E)
    = P
    ({ω : X (ω) ∈ E })
    . Explain why this yields a well defined probability measure on ℬ
    (ℝn)
    . This is called the distribution measure.
12. Let K ⊆ V where K is closed and V is open. Consider the following function.

    ( ) -----dist-x,V-C------- f (x) = dist(x,K )+ dist(x,V C)

    Explain why this function is continuous, equals 0 off V and equals 1 on K.

13. Let
    (Ω, ℱ)
    be a measurable space and let f : Ω → ℝⁿ be a measurable function. Then σ
    (f )
    denotes the smallest σ algebra such that f is measurable with respect to this σ algebra. Show that σ
    (f)
    =
    { − 1 n } f (E ) : E ∈ ℬ (ℝ )
    . More generally, you have a whole set of measurable functions S and σ
    (S )
    denotes the smallest σ algebra such that each function in S is measurable. If you have an increasing list S_t for t ∈ [0,∞), then σ
    (St)
    will be what is called a filtration. You have a σ algebra for each t ∈ [0,∞) and as t increases, these σ algebras get larger. This is an essential part of the construction which is used to show that Wiener process is a martingale. In fact the whole subject of martingales has to do with filtrations.
14. There is a monumentally important theorem called the Borel Cantelli lemma. It says the following. If you have a measure space
    (Ω,ℱ, μ)
    and if
    {Ei}
    ⊆ℱ is such that ∑ _i=1^∞μ
    (Ei)
    < ∞, then there exists a set N of measure 0 (μ
    (N )
    = 0) such that if ω
    ∈∕
    N, then ω is in only finitely many of the E_i. Hint: You might look at the set of all ω which are in infinitely many of the E_i. First explain why this set is of the form ∩_n=1^∞∪_k≥nE_k.
15. Let
    (Ω, ℱ,μ)
    be a measure space. A sequence of functions
    {fn}
    is said to converge in measure to a measurable function f if and only if for each ε > 0,

    nli→m∞ μ (ω : |fn(ω)− f (ω )| > ε) = 0

    Show that if this happens, then there exists a set of measure N such that if ω

    ∕∈
    N, then

    nl→im∞ fn(ω) = f (ω ).

    Also show that lim_n→∞f_n

    (ω )
    = f
    (ω)
    , then f_n converges in measure to f.