9.2. MEASURES AND THEIR PROPERTIES 241

function, being the composition of a continuous function, y→ d (xk,y)− d (x1,y) and ameasurable function, ω → f (ω) . Next let

A2 ≡{

ω /∈ A1 : d (x2, f (ω)) = mink≤n

d (xk, f (ω))

}and continue in this manner obtaining disjoint measurable sets, {Ak}n

k=1 such that for ω ∈Ak the best approximation to f (ω) from Dn is xk. Then fn (ω)≡ ∑

nk=1 xkXAk (ω) . Note

d ( fn+1 (ω) , f (ω)) = mink≤n+1

d (xk, f (ω))≤mink≤n

d (xk, f (ω)) = d ( fn (ω) , f (ω))

and so this verifies 9.4. It remains to verify 9.5.Let ε > 0 be given and pick ω ∈Ω. Then there exists xn ∈D such that d (xn, f (ω))< ε .

It follows from the construction that

d ( fn (ω) , f (ω))≤ d (xn, f (ω))< ε.

This proves the first half.Conversely, suppose the existence of the sequence of simple functions as described

above. Each fn is a measurable function because f−1n (U) = ∪{Ak : xk ∈U}. Therefore,

the conclusion that f is measurable follows from Theorem 9.1.2 on Page 237. ■Another useful observation is that the set where a sequence of measurable functions

converges is also a measurable set.

Proposition 9.1.8 Let { fn} be measurable with values in a complete normed vectorspace. Let A≡ {ω : { fn (ω)} converges} . Then A is measurable.

Proof: The set A is the same as the set on which { fn (ω)} is a Cauchy sequence. Thisset is

∩∞n=1∪∞

m=1∩p,q>m

[∥∥ fp (ω)− fq (ω)∥∥< 1

n

]which is a measurable set thanks to the measurability of each fn. ■

9.2 Measures and their PropertiesWhat is meant by a measure?

Definition 9.2.1 Let (Ω,F ) be a measurable space. Here F is a σ algebra of setsof Ω. Then µ : F → [0,∞] is called a measure if whenever {Fi}∞

i=1 is a sequence of disjointsets of F , it follows that

µ (∪∞i=1Fi) =

∞

∑i=1

µ (Ei)

Note that the series could equal ∞. If µ (Ω) < ∞, then µ is called a finite measure. Animportant case is when µ (Ω) = 1 when it is called a probability measure.

Note that µ ( /0) = µ ( /0∪ /0) = µ ( /0)+µ ( /0) and so µ ( /0) = 0.

Example 9.2.2 You could have P (N) = F and you could define µ (S) to be the numberof elements of S. This is called counting measure. It is left as an exercise to show that thisis a measure.