# Integral - Formal Definitions - Lebesgue Integral

Lebesgue Integral

It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann integrable, and so such limit theorems do not hold with the Riemann integral. Therefore it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated (Rudin 1987).

Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus Henri Lebesgue introduced the integral bearing his name, explaining this integral thus 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. This is the Riemann integral. But I can proceed differently. 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. This is my integral.

As Folland (1984, p. 56) puts it, "To compute the Riemann integral of f, one partitions the domain into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f". The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the Lebesgue measure μ(A) of an interval A = is its width, ba, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.

Using the "partitioning the range of f" philosophy, the integral of a non-negative function f : RR should be the sum over t of the areas between a thin horizontal strip between y = t and y = t + dt. This area is just μ{ x : f(x) > t} dt. Let f∗(t) = μ{ x : f(x) > t}. The Lebesgue integral of f is then defined by (Lieb & Loss 2001)

where the integral on the right is an ordinary improper Riemann integral (note that f∗ is a strictly decreasing positive function, and therefore has a well-defined improper Riemann integral). For a suitable class of functions (the measurable functions) this defines the Lebesgue integral.

A general measurable function f is Lebesgue integrable if the area between the graph of f and the x-axis is finite:

In that case, the integral is, as in the Riemannian case, the difference between the area above the x-axis and the area below the x-axis:

where

begin{align} f^+(x)&=max({f(x),0}) &=&begin{cases} f(x), & text{if } f(x) > 0, \ 0, & text{otherwise,} end{cases}\ f^-(x) &=max({-f(x),0})&=& begin{cases} -f(x), & text{if } f(x) < 0, \ 0, & text{otherwise.} end{cases} end{align}

### Other articles related to "integral, lebesgue integral, integrals, lebesgue":

Bochner Integral - Definition
... The Bochner integral is defined in much the same way as the Lebesgue integral ... bi ≠ 0, then the simple function is integrable, and the integral is then defined by exactly as it is for the ordinary Lebesgue integral ... of integrable simple functions sn such that where the integral on the left-hand side is an ordinary Lebesgue integral ...
Improper Integral - Types of Integrals
... From the point of view of calculus, the Riemann integral theory is usually assumed as the default theory ... In using improper integrals, it can matter which integration theory is in play ... For the Riemann integral (or the Darboux integral, which is equivalent to it), improper integration is necessary both for unbounded intervals (since one ...
Improper Integral - Improper Riemann Integrals and Lebesgue Integrals
... In some cases, the integral can be defined as an integral (a Lebesgue integral, for instance) without reference to the limit but cannot otherwise be ... In such cases, the improper Riemann integral allows one to calculate the Lebesgue integral of the function ... If a function f is Riemann integrable on for every b ≥ a, and the partial integrals are bounded as b → ∞, then the improper Riemann integrals both exist ...
Work - List of Publications of Ralph Henstock
... These were "On interval functions and their integrals" I (21, 1946) and II (23, 1948) "The efficiency of matrices for Taylor series" (22, 1947) "The efficiency of matrices for bounded sequences" (25, 1950) "The ... of generalized forms of the Ward, variational, Denjoy-Stieltjes, and Perron-Stieltjes integrals" ( 10, 1960) "N-variation and N-variational integrals of set functions" ( 11, 1961) "Definitions of Riemann ... On Ward’s Perron-Stieltjes integral, Canadian Journal of Mathematics 9 (1957) 96-109 ...
Hilbert Space - History
... was the observation, which arose during David Hilbert and Erhard Schmidt's study of integral equations, that two square-integrable real-valued functions f ... The second development was the Lebesgue integral, an alternative to the Riemann integral introduced by Henri Lebesgue in 1904 ... The Lebesgue integral made it possible to integrate a much broader class of functions ...

### Famous quotes containing the word integral:

... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.
Angelina Grimké (1805–1879)