Let m denote one dimensional Lebesgue measure. That is, it is the Lebesgue Stieltjes measure
which comes from the integrator function, F
(x)
= x. Also let the σ algebra of measurable sets
be denoted by ℱ. Recall this σ algebra contained the open sets. Also from the construction
given above,
m ([a,b]) = m ((a,b)) = b− a
Definition 9.1.1Let f be a function of n variables and consider the symbol
∫ ∫
⋅⋅⋅ f (x1,⋅⋅⋅,xn)dxi1 ⋅⋅⋅dxin. (9.1)
(9.1)
where
(i1,⋅⋅⋅,in)
is a permutation of the integers
{1,2,⋅⋅⋅,n }
. The symbol means to first dothe Lebesgue integral
∫
f (x ,⋅⋅⋅,x )dx
1 n i1
yielding a function of the other n − 1 variables given above. Then you do
∫ ( ∫ )
f (x ,⋅⋅⋅,x )dx dx
1 n i1 i2
and continue this way. The iterated integral is said to make sense if the process just describedmakes sense at each step. Thus, to make sense, it is required
xi1 → f (x1,⋅⋅⋅,xn)
can be integrated. Either the function has values in
[0,∞ ]
and is measurable or it is afunction in L1. Then it is required
∫
xi → f (x1,⋅⋅⋅,xn)dxi
2 1
can be integrated and so forth. The symbol in 9.1is called an iterated integral.
With the above explanation of iterated integrals, it is now time to define n dimensional
Lebesgue measure.