Very crudely and informally, the Riemann integral considers the mathematical limit of the areas of approximating "boxes" defined by intervals in the domain of the function, and the Lebesgue integral, by contrast, considers the limits of the "areas" defined under the curve over sets defined by intervals in the range of the function. Since the sets defined by these ranges can be significantly more general than the idea of two-dimensional rectangles, the Lebesque integral can be defined over a significantly more general range of functions.
This notion of integral
is closely related to that of a Lebesgue measure. Indeed, the
Lebesgue integral
In modern mathematics, the Lebesgue integral (and not the Riemman
integral), is the most used notion of integral. This is because of
some technical advantages of the Lebesgue integral that we examine
bellow, by comparing the two notions of integral.
1) The class of Lebesgue integrable functions is much richer than that
of Riemman integrable functions, as the following example shows:
For the Lebesgue integral, hoewever the Dirichlet Function is
integrable and its integral is zero since, the Lebesgue measure
of the set of rational numbers in [0,1] is zero.
2) The key property of the Lebesgue integral is that the the passage
to the limit in the integral can be done under rather weak
conditions. For example, the monotone convergence theorem asserts
that for a non decreasing sequence of non negative measurable functions is always true that
Informal introduction
Detailed treatment
for a non negative Lebesgue-measurable function can be defined
as the (bidimensional) Lebesgue measure of the set
between the x-axes and the graph of f(several equivalent definitions are possible, this one resembles the geometric interpretation
of the integral in elementary [calculus]). For a function of arbitrary sign,
we can use the decomposition
where
is the positive part, and
is the negative part. Then, we may define
if both the integrals of and are finite. In that case, the function f is
said to be Lebesgue integrable.
interval, any rectangle used will have
height 1 (because all rectangles contain rational points) and in the lower sum, any rectangle used will have height 0 (because all rectangles contain irrational points.) Hence the lower sum is 0 and the upper sum is 1.
This is closely related to the fact that the Lebesgue measure is
contably additive.
However, this theorem does not hold for the Riemman integral. In order to see why this is so, let ak be an enumeration of all the rational numbers in [0,1] (they are countable so this can be done.) Then let gk be the function which is 1 on ak and 0 everywhere else. Lastly let fk = g1 + g2 + ... + gk. Then fk is zero everywhere except on a finite set of points, hence its Riemann integral is zero. The sequence fk is also clearly non-negative and monotonously increasing to H(x), but H(x) isn't Riemann integrable.
3) The Riemann integral can only integrate functions on an interval. The simplest extension is to define ∫− ∞∞f(x) dx by the limit of ∫−aaf(x) dx as a goes to +∞. However, this breaks translation invariance: if f and g are zero outside some interval [a, b] and are Riemann integrable, and if f(x) = g(x + y) for some y, then ∫ f = ∫ g. However, with this definition of the improper integral (this definition is sometimes called the improper Cauchy principal value about zero), the functions f(x) = (1 if x > 0, −1 otherwise) and g(x) = (1 if x > 1, −1 otherwise) are translations of one another, but their improper integrals are different. (∫ f = 0 but ∫ g = − 2.)
On the other hand, for the Lebesgue Integral, the definition of integral over the whole line presents no difficulties (in fact there is no need to distinguish between proper and improper integrals). Morover, the concept of Lebesgue integral can be generalized to measurable functions defined on arbitrary measure spaces.
The Lebesgue integral is an important tool in all the branches of mathematics that are related to Analysis: for example in harmonic analysis, in functional analysis where it plays a role in the definition of Lp spaces and in probability theory.