33.5. LEBESGUE MEASURE AND LINEAR MAPS 661

This will suffice for this book. Actually, you use the completion of this measure spaceand this completion is Lebesgue measure. Writing such a measurable function as a differ-ence between positive and negative parts, gives the following corollary.

Corollary 33.4.3 In the context of the above Proposition 33.4.2, if f ∈ L1 (Rp) , thenfor any permutation (i1, · · · , ip) of {1, · · · , p} it follows∫

Rpf dmp =

∫· · ·∫

f (x1, · · · ,xp)dxi1 · · ·dxip (33.4)

Note that this implies that if∫· · ·∫ ∣∣ f (x1, · · · ,xp)

∣∣dxi1 · · ·dxip <∞, the integration takenin any order, then 33.4 holds for all permutations.

The next big theorem about the integral is the change of variables formula. RecallLemma 32.1.6.

Lemma 33.4.4 Every open set in Rp is the countable disjoint union of half open boxesof the form

p

∏i=1

(ai,ai +2−k]

where ai = l2−k for some integers, l,k where k ≥ m. If Bm denotes this collection of halfopen boxes, then every box of Bm+1 is contained in a box of Bm or equals a box of Bm.

33.5 Lebesgue Measure and Linear MapsLemma 33.5.1 Let A : Rp → Rp be linear and invertible. Then A maps open sets toopen sets.

Proof: This follows from the observation that if B is any linear transformation, then B iscontinuous. Indeed, it is realized by matrix multiplication and so it is clear that if xn → x,then Bxn → Bx. Now it follows that A−1 is continuous. Let U be open. Let y ∈ A(U) .Then is y an interior point of A(U)? if not, there exists yn → y where yn /∈ A(U). But thenA−1yn → A−1y ∈U. Since U is open, A−1yn ∈U for all n large enough and so yn ∈ A(U)after all. Thus y is an interior point of A(U) showing that A(U) is open.

Corollary 33.5.2 Let A : Rp → Rp be linear and invertible. Then A maps Borel sets toBorel sets.

Proof: Let the pi system be K the open sets. Then let G be those Borel sets E such thatA(E) is Borel. Then it is clear that G contains K and is closed with respect to complementsand countable disjoint unions. By Dynkin’s lemma, G = B (Rp) = σ (K ). This lastequality holds by definition of the Borel sets B (Rp).

From Linear algebra, Chapter 18 the chapter on row operations and elementary matri-ces, if A is such an invertible linear transformation, it is the composition of finitely manyinvertible linear transformations which are of the following form.(

x1 · · · xr · · · xs · · · xp)T

→(

x1 · · · xr · · · xs · · · xp)T

(x1 · · · xr · · · xp

)T →(

x1 · · · cxr · · · xp)T

,c ̸= 0