258 CHAPTER 11. ABSTRACT MEASURE AND INTEGRATION

Therefore, by Fatou’s lemma,∫Ω

| f |dµ ≤ lim infn→∞

∫| fn|dµ ≤M,

showing that f ∈ L1 as hoped.Now S∪{ f} is uniformly integrable so there exists δ 1 > 0 such that if µ (E) < δ 1,

then∫

E |g|dµ < ε/3 for all g ∈ S∪{ f}. By Egoroff’s theorem, there exists a set, F withµ (F) < δ 1 such that fn converges uniformly to f on FC. Therefore, there exists N suchthat if n > N, then ∫

FC| f − fn|dµ <

ε

3.

It follows that for n > N,∫Ω

| f − fn|dµ ≤∫

FC| f − fn|dµ +

∫F| f |dµ +

∫F| fn|dµ

<ε

3+

ε

3+

ε

3= ε,

which verifies 11.5.25.

11.6 Measures and RegularityIt is often the case that Ω has more going on than to simply be a set. In particular, it is oftenthe case that Ω is some sort of topological space, often a metric space. In this case, it isusually if not always the case that the open sets will be in the σ algebra of measurable sets.This leads to the following definition.

Definition 11.6.1 A Polish space is a complete separable metric space. For a Polish spaceE or more generally a metric space or even a general topological space, B (E) denotes theBorel sets of E. This is defined to be the smallest σ algebra which contains the open sets.Thus it contains all open sets and closed sets and compact sets and many others.

Don’t ever try to describe a generic Borel set. Always work with the definition that itis the smallest σ algebra containing the open sets. Attempts to give an explicit descriptionof a “typical” Borel set tend to lead nowhere because there are so many things which canbe done.You can take countable unions and complements and then countable intersectionsof what you get and then another countable union followed by complements and on andon. You just can’t get a good useable description in this way. However, it is easy to see

that something like(∩∞

i=1∪∞j=i E j

)Cis a Borel set if the E j are. This is useful. This said,

you can look at Hewitt and Stromberg in their discussion of why there are more Lebesguemeasurable sets than Borel measurable sets to see the kind of technicalities which result bydescribing Borel sets.

For example, R is a Polish space as is any separable Banach space. Amazing thingscan be said about finite measures on the Borel sets of a Polish space. First the case of afinite measure on a metric space will be considered.