32.2. SIMPLE FUNCTIONS, σ ALGEBRAS, MEASURABILITY 633

where µ (Ei) denotes the measure of Ei. This is a more general concept than length. Itcould refer to probability that a random vector has values in the event Ei ⊆Rp for example.

We would like to be able to measure all subsets of a given set Ω but it turns out thatthis is usually impossible to include along with all of the following definition. This willbecome clear a little later in the discussion of outer measures. However, the notion of a σ

algebra turns out to be the ideal thing for a theory of integration.

Definition 32.2.1 Let Ω be a nonempty set. A σ algebra F is a set whose elementsare subsets of Ω which satisfies the following.

1. If Ei ∈ F , for i = 1,2, · · · , then ∪∞i=1Ei ∈ F .

2. If E ∈ F , then EC ≡ Ω\E ∈ F

3. /0,Ω are both in F

µ : F → [0,∞] is called a measure if whenever Ei ∈ F and Ei ∩E j = /0 for all i ̸= j,then

µ (∪∞i=1Ei) =

∞

∑i=1

µ (Ei)

that sum is defined as supn ∑ni=1 µ (Ei) . It could be a real number or +∞. Such a pair

(Ω,F ) is called a measurable space. If you add in µ, written as (Ω,F ,µ) , it is called ameasure space.

Of course our main interest is where Ω is a nonempty subset of R or Rp and the measureµ is something to do with length or p dimensional volume, returning the length for aninterval or volume of a p dimensional box, but it is no more trouble to present this in thegenerality just described and such a generalization is essential to understand if you wantto study mathematical statistics or probability. Surely the study of the integral should leadsomewhere.

Observation 32.2.2 If (Ω,F ) is a measurable space and Ei ∈ F , then ∩∞i=1Ei ∈ F .

This is because Ei ∈ F and by DeMorgan’s laws,

∩∞i=1Ei =

(∪∞

i=1ECi)C ∈ F since each EC

i ∈ F

Measures have the following fundamental property.

Lemma 32.2.3 If µ is a measure and Fi ∈ F , then µ (∪∞i=1Fi) ≤ ∑

∞i=1 µ (Fi). Also if

Fn ∈ F and Fn ⊆ Fn+1 for all n, then if F = ∪nFn,

µ (F) = limn→∞

µ (Fn)

Symbolically, if Fn ↑ F, then µ (Fn) ↑ µ (F). If Fn ⊇ Fn+1 for all n, then if µ (F1) < ∞ andF = ∩nFn, then

µ (F) = limn→∞

µ (Fn)

Symbolically, if µ (F1)< ∞ and Fn ↓ F, then µ (Fn) ↓ µ (F).