First a set theoretic definition: Let E be a measurable set (in terms of Lebesgue measure) in Rn. A dissection of this set is a collection C that consists of disjoint, measurable sets whose union is E.
Let f be an extended real function defined on E, and let C be a dissection of E. For every set Cr in C, let
B (f) = sup f(x)
b (f) = inf f(x)
h (f) = sup f(x) = max(|b (f)|, |B (f)|)
r x∈C r r
Let us say that if hr(f) = ∞ and m(Cr) = 0, or vice-versa, then we take hr(f)m(Cr) = 0. A dissection C such that
Σ h (f)m(C ) < ∞
is called admissible. Let the collection of all admissible dissections for a function f over a measurable set E be denoted A(f, E) or just A, if the function and set are understood.
For any dissection C in A, we define the sum
S (f) = Σ B (f) m(C )
C r r
to be the upper approximating sum and similarly
s (f) = Σ b (f) m(C )
C r r
for the lower approximating sum of f over E corresponding to C.
We now define the upper integral with respect to Lebesgue measure as
∫ f = inf S (f)
E C∈A C
and the lower integral as
∫ f = sup s (f)
*E C∈A C
We say that a function f is Lebesgue integrable over a set E if its upper and lower integrals over E are equal, and the value of both is defined to be the Lebesgue integral of f over E.
Clearly, any function is integrable over a set of measure zero, and its integral is zero. Also, any function that is zero almost everywhere (such as the Dirichlet function) is integrable over any set, and its integral is also zero.
A necessary and sufficient condition that f should be integrable over E is that for any ε > 0 there should exist an admissible dissection C of E where SC - sC < ε.
There other ways of defining the Lebesgue integral, such as the Daniell integral, which are equivalent.
This is clearly a much more complicated definition of the integral than the Riemann integral, but it has many benefits over that definition, chief of which is that the set of functions integrable in this way forms a Banach space given the natural norm, the Riesz-Fischer Theorem establishes this important result. Also many other functions whose integral does not exist in the Riemann definition have Lebesgue Integrals.