Chapter 13

Lebesgue Measure13.1 Basic Properties

Definition 13.1.1 Define the following positive linear functional for f ∈Cc (Rn) .

Λ f ≡∫

∞

−∞

· · ·∫

∞

−∞

f (x)dx1 · · ·dxn.

Then the measure representing this functional is Lebesgue measure.

The following lemma will help in understanding Lebesgue measure.

Lemma 13.1.2 Every open set in Rn is the countable disjoint union of half open boxes ofthe form

n

∏i=1

(ai,ai +2−k]

where ai = l2−k for some integers, l,k. The sides of these boxes are of equal length. Onecould also have half open boxes of the form

n

∏i=1

[ai,ai +2−k)

and the conclusion would be unchanged.

Proof: Let

Ck = {All half open boxesn

∏i=1

(ai,ai +2−k] where

ai = l2−k for some integer l.}

Thus Ck consists of a countable disjoint collection of boxes whose union is Rn. This issometimes called a tiling of Rn. Think of tiles on the floor of a bathroom and you will getthe idea. Note that each box has diameter no larger than 2−k√n. This is because if

x,y ∈n

∏i=1

(ai,ai +2−k],

then |xi− yi| ≤ 2−k. Therefore,

|x−y| ≤

(n

∑i=1

(2−k)2)1/2

= 2−k√n.

Let U be open and let B1 ≡ all sets of C1 which are contained in U . If B1, · · · ,Bk havebeen chosen, Bk+1 ≡ all sets of Ck+1 contained in

U \∪(∪k

i=1Bi

).

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