Kenneth Kuttler

9.9. FUBINI’S THEOREM AN INTRODUCTION 213

that iterated integrals are what occur naturally in many situations, and each integral in aniterated integral is a one dimensional notion, so it is natural to consider the interchange ofiterated integrals in a book on single variable calculus. All of this depends on the theoremsabout continuous functions defined on a subset of Fp. In the case considered here, p = 2.

The following theorem is just like an earlier one for functions of one variable.

Theorem 9.9.1 Let f be either decreasing or increasing and let g be continuous on[a,b]. Then there exists c ∈ [a,b] such that

∫ ba gd f = g(c)( f (b)− f (c)).

Proof: If f is constant, there is nothing to prove so assume f (b)> f (a) or f (a)> f (b).Let M ≡ max{g(x) : x ∈ [a,b]} ,m ≡ min{g(x) : x ∈ [a,b]}. Then in a Riemann sum for∫ b

a gd f , if g is replaced by M, the resulting Riemann sum will increase and if it is replacedwith m, the resulting sum will decrease. Therefore, in case f is increasing,

m( f (b)− f (a)) =∫ b

amd f ≤

∫ b

agd f ≤

∫ b

aMd f = M ( f (b)− f (a))

and so m≤ 1f (b)− f (a)

∫ ba gd f ≤M. In case f is decreasing,

m( f (b)− f (a)) =∫ b

amd f ≥

∫ b

agd f ≥

∫ b

aMd f = M ( f (b)− f (a))

and so m ≤ 1f (b)− f (a)

∫ ba gd f ≤ M.Therefore, by the intermediate value theorem, there is

c ∈ [a,b] such that 1f (b)− f (a)

∫ ba gd f = g(c) .

Lemma 9.9.2 Let f : [a,b]× [c,d]→ R be continuous at every point so it is uniformlycontinuous. Let α,β be increasing on [a,b] , [c,d] respectively. Then

x→∫ d

cf (x,y)dβ (y) , y→

∫ b

af (x,y)dα (x)

are both continuous functions.

Proof: Consider the first. The other is exactly similar.∣∣∣∣∫ d

cf (x,y)dβ (y)−

∫ d

cf (x̂,y)dβ (y)

∣∣∣∣= ∣∣∣∣∫ d

c( f (x,y)− f (x̂,y))dβ (y)

∣∣∣∣≤∫ d

c| f (x,y)− f (x̂,y)|dβ (y)

But by uniform continuity, if |x− x̂| is small enough, then | f (x,y)− f (x̂,y)|< ε and so theintegral in the above is no larger than ε (β (d)−β (c)). Since ε is arbitrary, this shows theclaim.

Note that, since these are continuous functions, it follows from Theorem 9.3.7 that itmakes perfect sense to write the iterated integrals∫ b

∫ d

cf (x,y)dβ (y)dα (x) ,

∫ d

∫ b

af (x,y)dα (x)dβ (y)

Of course the burning question is whether these two numbers are equal. This is the nexttheorem.

9.9. FUBINI’'S THEOREM AN INTRODUCTION 213that iterated integrals are what occur naturally in many situations, and each integral in aniterated integral is a one dimensional notion, so it is natural to consider the interchange ofiterated integrals in a book on single variable calculus. All of this depends on the theoremsabout continuous functions defined on a subset of F”. In the case considered here, p = 2.The following theorem is just like an earlier one for functions of one variable.Theorem 9.9.1 Lez f be either decreasing or increasing and let g be continuous on[a,b]. Then there exists c € [a,b] such that fe gdf = g(c)(f(b)—f (c)).Proof: If f is constant, there is nothing to prove so assume f (b) > f (a) or f (a) > f (bd).Let M = max {g (x) : x € |[a,b]} ,m = min {g (x) :x € [a,b]}. Then in a Riemann sum forf ° gdf, if g is replaced by M, the resulting Riemann sum will increase and if it is replacedwith m, the resulting sum will decrease. Therefore, in case f is increasing,m(f(o)—s(a)) = [map < [gap < [Map =M(r)—F(@)and som < f° gdf <M. Incase f is decreasing,ORIOREm(s(o)— F(a) = [mar [gar [mar =Mcr)—s(a))and som < on b gdf <M.Therefore, by the intermediate value theorem, there isa)c€ |a, b] such that pt w Jn sdf = g(c).Lemma 9.9.2 Let f : [a,b] x [c,d] + R be continuous at every point so it is uniformlycontinuous. Let a, B be increasing on [a,b] ,|c,d] respectively. Thenx ['ree)4B0).9> [pesadoare both continuous functions.Proof: Consider the first. The other is exactly similar.['ronap0)- [' r06.48.0)] =< [Wrox 08 9))4B 6)But by uniform continuity, if |x —£| is small enough, then | f (x,y) — f (%,y)| < € and so theintegral in the above is no larger than € (B (d) — B (c)). Since € is arbitrary, this shows theclaim. §Note that, since these are continuous functions, it follows from Theorem 9.3.7 that itmakes perfect sense to write the iterated integralsPps (x,y) dB (vy) da (x 1 [ [ resaate ) dB (y)Of course the burning question is whether these two numbers are equal. This is the nexttheorem.[e914 6)