- Show carefully that if S is a set whose elements are σ algebras which are subsets
of P, then ∩S is also a σ algebra. Now let G⊆Psatisfy 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.
- The Borel sets of a metric space are the sets in the smallest σ algebra which contains the open sets. These sets are denoted as ℬ. Thus ℬ= σwhere σsimply means the smallest σ algebra which contains ℱ. Show ℬ= σwhere P consists of the half open rectangles which are of the form ∏
_{i=1}^{n}[a_{i},b_{i}). - Recall that f : → X where X is a metric space is measurable means f
^{−1}∈ ℱ. Show that if E is any set in ℬ, then f^{−1}∈ ℱ. Thus, inverse images of Borel sets are measurable. Next consider f :→ X being measurable and g : X → Y is Borel measurable, meaning that g^{−1}∈ ℬ. Explain why g ∘ f is measurable. Hint: You know that^{−1}= f^{−1}. 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 course. - If you have X
_{i}is a metric space, let X = ∏_{i=1}^{n}X_{i}with the metricYou considered this in an earlier problem. Show that any set of the form

is a Borel set. That is, the product of Borel sets is Borel. Hint: You might consider the continuous functions π

_{i}: ∏_{j=1}^{n}X_{j}→ X_{i}which are the projection maps. Thus π_{i}≡ x_{i}. Then π_{i}^{−1}would have to be Borel measurable whenever E_{i}∈ℬ. Explain why. You know π_{i}is continuous. Why would π_{i}^{−1}be a Borel set? Then you might argue that ∏_{i=1}^{n}E_{i}= ∩_{i=1}^{n}π_{i}^{−1}. - You have two finite measures defined on ℬμ,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.
- Show that is a measure space where μequals the number of elements of S. You need to verify that if the sets E
_{i}are disjoint, then μ= ∑_{i=1}^{∞}μ. - 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= 1 if A is uncountable and μ= 0 if A is countable. Show that μ is a measure. - Let μ= 1 if 0 ∈ E and μ= 0 if 0E. Show this is a measure on P.
- Give an example of a measure μ and a measure space and a decreasing sequence of
measurable sets such that lim
_{n→∞}μ≠μ. - If you have a finite measure μ on ℬwhere X is a metric space, and if F ∈ℬ, 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= 0.
- You have a measure space where P is a probability measure on ℱ. Then you also have a measurable function X : Ω → Z where Z is some metric space. Thus X
^{−1}∈ℱ whenever U is open. Now define a measure on ℬdenoted by λ_{X}and defined by λ_{X}= P. Explain why this yields a well defined probability measure on ℬwhich is regular. This is called the distribution measure. - Let K ⊆ V where K is closed and V is open. Consider the following function.
Explain why this function is continuous, equals 0 off V and equals 1 on K.

- Let be a measurable space and let f : Ω → X be a measurable function. Then σdenotes the smallest σ algebra such that f is measurable with respect to this σ algebra. Show that σ=. More generally, you have a whole set of measurable functions S and σdenotes the smallest σ algebra such that each function in S is measurable. If you have an increasing list S
_{t}for t ∈ [0,∞), then σwill 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. - There is a monumentally important theorem called the Borel Cantelli lemma. It says the
following. If you have a measure space and if⊆ℱ is such that ∑
_{i=1}^{∞}μ< ∞, then there exists a set N of measure 0 (μ= 0) such that if ωN, 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≥n}E_{k}. - Let be a measure space. A sequence of functionsis said to converge in measure to a measurable function f if and only if for each ε > 0,
Show that if this happens, then there exists a set of measure N such that if ω

N, thenAlso show that lim

_{n→∞}f_{n}= f, then f_{n}converges in measure to f. - Let X,Y be separable metric spaces. Then X × Y can also be considered as a metric
space with the metric
Verify this. Then show that if K consists of sets A×B where A,B are Borel sets in X and Y respectively, then it follows that σ

= ℬ, the Borel sets from X ×Y . Extend to the Cartesian product ∏_{i}X_{i}of finitely many separable metric spaces.

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