662 CHAPTER 33. THE LEBESGUE MEASURE AND INTEGRAL IN Rp

(x1 · · · xr · · · xs · · · xp

)T

→(

x1 · · · xr · · · xs + xr · · · xp)T

where these are the actions obtained by multiplication by elementary matrices. Denotethese special linear transformations by E (r ↔ s) ,E (cr) ,E (s → s+ r) .

Let R = ∏pi=1 (ai,bi) . Then it is easily seen that

mp (E (r ↔ s)(R)) = mp (R) = |det(E (r ↔ s))|mp (R)

mp (E (cr)(R)) = |c|mp (R) = |det(E (cr))|mp (R)

The other linear transformation which represents a sheer is a little harder. However,

mp (E (s → s+ r)(R)) =∫

E(s→s+r)(R)dmp

=∫R· · ·∫R

∫R

∫R

XE(s→s+r)(R)dxsdxrdxp1 · · ·dxpp−2

Now recall Theorem 33.3.2 which says you can integrate using the usual Riemannintegral when the function involved is continuous. Thus the above becomes∫ bpp−2

app−2

· · ·∫ bp1

ap1

∫ br

ar

∫ bs+xr

as+xr

dxsdxrdxp1 · · ·dxpp−2

= mp (R) = |det(E (s → s+ r))|mp (R)

Recall that when a row (column) is added to another row (column), the determinant of theresulting matrix is unchanged.

Lemma 33.5.3 Let L be any of the above elementary linear transformations. Then

mp (L(F)) = |det(L)|mp (F)

for any Borel set F. Also L(F) is Borel if F is Borel.

Proof: Let Rk = ∏pi=1 (−k,k) . Let G be those Borel sets F such that

mp (L(F ∩Rk)) = |det(L)|mp (F ∩Rk) (33.5)

Letting K be the open boxes, it follows from the above discussion that the pi system Kis in G . It is also obvious that if Fi ∈ G the Fi being disjoint, then

mp (L(∪∞i=1Fi ∩Rk)) =

∞

∑i=1

mp (L(Fi ∩Rk)) = |det(L)|∞

∑i=1

mp (Fi ∩Rk)

= |det(L)|mp (∪∞i=1Fi ∩Rk)

Thus G is closed with respect to countable disjoint unions. If F ∈ G then

mp(L(FC ∩Rk

))+mp (L(F ∩Rk)) = mp (L(Rk))

mp(L(FC ∩Rk

))+ |det(L)|mp (F ∩Rk) = |det(L)|mp (Rk)

mp(L(FC ∩Rk

))= |det(L)|mp (Rk)−|det(L)|mp (F ∩Rk)

= |det(L)|mp(FC ∩Rk

)It follows that G is closed with respect to complements also. Therefore, G = σ (K ) =B (Rp). Now let k → ∞ in 33.5 to obtain the desired conclusion.