Lebesgue integral

It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral.

The integral of a positive real function f between boundaries a and b can be interpreted as the area under the graph of f, between a and b.

Graphs like the one of the latter, raise the question: for which class of functions does "area under the curve" make sense?

As part of a general movement toward rigor in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on a firm foundation.

Riemann's definition starts with the construction of a sequence of easily calculated areas that converge to the integral of a given function.

For this reason, the Lebesgue definition makes it possible to calculate integrals for a broader class of functions.

Lebesgue summarized his approach to integration in a letter to Paul Montel: I have to pay a certain sum, which I have collected in my pocket.

I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum.

After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor.

Folland (1999) summarizes the difference between the Riemann and Lebesgue approaches thus: "to compute the Riemann integral of f, one partitions the domain [a, b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f." For the Riemann integral, the domain is partitioned into intervals, and bars are constructed to meet the height of the graph.

The slabs viewpoint makes it easy to define the Lebesgue integral, in terms of basic calculus.

is a (Lebesgue measurable) function, taking non-negative values (possibly including

Most textbooks, however, emphasize the simple functions viewpoint, because it is then more straightforward to prove the basic theorems about the Lebesgue integral.

This suggests that picking out a suitable class of measurable subsets is an essential prerequisite.

Riemann could only use planar rectangles to approximate the area under the curve, because there was no adequate theory for measuring more general sets.

In the development of the theory in most modern textbooks (after 1950), the approach to measure and integration is axiomatic.

This means that a measure is any function μ defined on a certain class X of subsets of a set E, which satisfies a certain list of properties.

The integral of a simple function is found by summing, over these (not necessarily connected) subsets of the domain, the product of the measure of the subset and its image under the simple function (the lower bound of the corresponding layer); intuitively, this product is the sum of the areas of all bars of the same height.

Some care is needed when defining the integral of a real-valued simple function, to avoid the undefined expression ∞ − ∞: one assumes that the representation

Then the above formula for the integral of f makes sense, and the result does not depend upon the particular representation of f satisfying the assumptions.

We need to show this integral coincides with the preceding one, defined on the set of simple functions, when E is a segment [a, b].

It is often useful to have a particular sequence of simple functions that approximates the Lebesgue integral well (analogously to a Riemann sum).

Each gk is non-negative, and this sequence of functions is monotonically increasing, but its limit as k → ∞ is 1Q, which is not Riemann integrable.

It can however be extended to unbounded intervals by taking limits, so long as this doesn't yield an answer such as ∞ − ∞.

The Riemann integral is inextricably linked to the order structure of the real line.

[9] Necessary and sufficient conditions for the interchange of limits and integrals were proved by Cafiero,[10][11][12][13] generalizing earlier work of Renato Caccioppoli, Vladimir Dubrovskii, and Gaetano Fichera.

There is also an alternative approach to developing the theory of integration via methods of functional analysis.

The Riemann integral exists for any continuous function f of compact support defined on Rn (or a fixed open subset).

Furthermore, the Riemann integral ∫ is a uniformly continuous functional with respect to the norm on Cc, which is dense in L1.

More precisely, the compactly supported functions form a vector space that carries a natural topology, and a (Radon) measure is defined as a continuous linear functional on this space.