Kenneth Kuttler

660 CHAPTER 33. THE LEBESGUE MEASURE AND INTEGRAL IN Rp

given above,m([a,b]) = m((a,b)) = b−a

Definition 33.4.1 Let f be a function of p variables and consider the s ymbol∫· · ·∫

f (x1, · · · ,xp)dxi1 · · ·dxip . (33.2)

where (i1, · · · , ip) is a permutation of the integers {1,2, · · · , p} . The symbol means to firstdo the Lebesgue integral ∫

f (x1, · · · ,xp)dxi1

yielding a function of the other p−1 variables given above. Then you do∫ (∫f (x1, · · · ,xp)dxi1

)dxi2

and continue this way. The iterated integral is said to make sense if the process just de-scribed makes sense at each step. Thus, to make sense, it is required

xi1 → f (x1, · · · ,xp)

can be integrated. Either the function has values in [0,∞] and is measurable or it is afunction in L1. Then it is required

xi2 →∫

f (x1, · · · ,xp)dxi1

can be integrated and so forth. The symbol in 33.2 is called an iterated integral.

With the above explanation of iterated integrals, it is now time to define p dimensionalLebesgue measure.

33.4.2 p Dimensional Lebesgue Measure and IntegralsConsider (R,F ,m) the measure space corresponding to one dimensional Lebesgue mea-sure. Then from Proposition 32.12.2, we obtain the existence of p dimensional Lebesguemeasure.

Proposition 33.4.2 There exists a measure mp defined on F p such that if f : Rp →[0,∞) is measurable with respect to F p then for any permutation (i1, · · · , ip) of {1, · · · , p}it follows ∫

Rpf dmp =

∫· · ·∫

f (x1, · · · ,xp)dxi1 · · ·dxip (33.3)

In particular, this implies that if Ai is a Borel set for each i = 1, · · · , p then

∏i=1

m(Ai) .

and all such ∏pi=1 Ai is in F p.

660 CHAPTER 33. THE LEBESGUE MEASURE AND INTEGRAL IN R?given above,m/(|a,b]) =m((a,b)) =b—aDefinition 33.4.1 Le f be a function of p variables and consider the s ymbolJ fr X10°+ Xp) AX, AX. (33.2)where (i1,-+- ,ip) is a permutation of the integers {1,2,--- , p}. The symbol means to firstdo the Lebesgue integral[ri X15" rg p) Axi,yielding a function of the other p—1 variables given above. Then you do[ (fre Xpyrt x ip) ds) diand continue this way. The iterated integral is said to make sense if the process just de-scribed makes sense at each step. Thus, to make sense, it is requiredXi, > f (X10 Xp)can be integrated. Either the function has values in {0,°] and is measurable or it is afunction in L'. Then it is requiredXin > [fe Xp) Axi,can be integrated and so forth. The symbol in 33.2 is called an iterated integral.With the above explanation of iterated integrals, it is now time to define p dimensionalLebesgue measure.33.4.2. p Dimensional Lebesgue Measure and IntegralsConsider (R, #,m) the measure space corresponding to one dimensional Lebesgue mea-sure. Then from Proposition 32.12.2, we obtain the existence of p dimensional Lebesguemeasure.Proposition 33.4.2 There exists a measure mp defined on F” such that if f :R? >[0,c°) is measurable with respect to ¥” then for any permutation (i1,--- ,ip) of {1,--- ,p}it follows| fadmy = fF Xy,01+ x p) AX), ** -dxi, (33.3)In particular, this implies that if A; is a Borel set for eachi=1,--- , p thenP PMp (I) = []m(ai)and all such []}_, Aj is in F?.