520 CHAPTER 19. HAUSDORFF SPACES AND MEASURES
Proof: Let spt( f ) ⊆ [−R,R]n. It clearly suffices to show this for n = 2. Then by thedefinition of the Riemann integral,
∫ R−R∫ R−R f (x,y)dxdy =∫ R
−R∑
i
∫ xi+1
xi
f (s,y)dsdy =n−1
∑j=0
n−1
∑i=0
∫ yi+1
yi
∫ xi+1
xi
f (s, t)dsdt
=n−1
∑j=0
n−1
∑i=0
∫ yi+1
yi
f (si, t)(xi+1− xi)dt
=n−1
∑j=0
n−1
∑i=0
f (si, t j)(xi+1− xi)(y j+1− y j
)=
n−1
∑i=0
n−1
∑j=0
f (si, t j)(xi+1− xi)(y j+1− y j
)where−R = x0 < x1 < · · ·< xn = R is a uniform partition of [−R,R] with the yi also givinga uniform partition of [−R,R]. Similar reasoning implies∫ R
−R
∫ R
−Rf (x,y)dydx =
n−1
∑i=0
n−1
∑j=0
f (ŝi, t̂ j)(xi+1− xi)(y j+1− y j
).
Now (si, t j) ,(ŝi, t̂ j) are both in [xi,xi+1]×[y j,y j+1
]. Thus, by uniform continuity, if n is
large enough, ∣∣ f (si, t j)− f (ŝi, t̂ j)∣∣< ε
4R2
Then it follows that∣∣∣∫ R−R∫ R−R f (x,y)dxdy−
∫ R−R∫ R−R f (x,y)dydx
∣∣∣≤n−1
∑i=0
n−1
∑j=0
∣∣ f (ŝi, t̂ j)− f (si, t j)∣∣(xi+1− xi)
(y j+1− y j
)≤
n−1
∑i=0
n−1
∑j=0
ε
4R2 (xi+1− xi)(y j+1− y j
)= ε
Since ε is arbitrary, this shows that the two iterated integrals are the same. In case n> 2, youcan do exactly the same argument using the mean value theorem for integrals and obtainthe same result by a similar argument, or you could use this result on pairs of integrals. ■
19.8 Slicing MeasuresI saw this material first in the book [17]. It can be presented as an application of the theoryof differentiation of Radon measures and the Riesz representation theorem for positivelinear functionals. It is an amazing theorem and can be used to understand conditionalprobability However, here I will obtain it from Theorem 10.14.12.
Theorem 19.8.1 Let µ be a finite Radon measure on Rn+m defined on a σ algebra,F . Then there exists a unique finite Radon measure α, defined on a σ algebra S , of setsof Rn which satisfies
α (E) = µ (E×Rm) (19.12)
for all E Borel. There also exists a Borel set of α measure zero N, such that for each x /∈N,there exists a Radon probability measure νx such that if f is a nonnegative µ measurablefunction or a µ measurable function in L1 (µ),
y→ f (x,y) is νx measurable α a.e.