Lebesgue integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the $X$ axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.

Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, one might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the necessary abstractions for this.

The Lebesgue integral plays an important role in probability theory, real analysis, and many other fields in mathematics. It is named after Henri Lebesgue (1875–1941), who introduced the integral (Lebesgue 1904). It is also a pivotal part of the axiomatic theory of probability.

The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or the specific case of integration of a function defined on a sub-domain of the real line with respect to the Lebesgue measure.

$1_{\mathbf {Q} }$ is not Riemann-integrable on $[ 0, 1]$ : No matter how the set $[ 0, 1]$ is partitioned into subintervals, each partition contains at least one rational and at least one irrational number, because rationals and irrationals are both dense in the reals. Thus the upper are all one, and the lower Darboux sums are all zero.

Darboux sums

$1_{\mathbf {Q} }$ is Lebesgue-integrable on $[ 0, 1]$ using the : Indeed, it is the indicator function of the rationals so by definition
$\int _{[0,1]}1_{\mathbf {Q} }\,d\mu =\mu (\mathbf {Q} \cap [0,1])=0,$ because $Q$ is countable.

Lebesgue measure

Consider the indicator function of the rational numbers, $1 Q$ , also known as the Dirichlet function. This function is nowhere continuous.

If $f$ and $g$ are non-negative measurable functions (possibly assuming the value $+\infty$ ) such that $f = g$ almost everywhere, then
$\int f\,d\mu =\int g\,d\mu .$ To wit, the integral respects the equivalence relation of almost-everywhere equality.

If $f$ and $g$ are functions such that $f = g$ almost everywhere, then $f$ is Lebesgue integrable if and only if $g$ is Lebesgue integrable, and the integrals of $f$ and $g$ are the same if they exist.

: If $f$ and $g$ are Lebesgue integrable functions and $a$ and $b$ are real numbers, then $af + bg$ is Lebesgue integrable and
$\int (af+bg)\,d\mu =a\int f\,d\mu +b\int g\,d\mu .$

Linearity

: If $f \leq g$ , then
$\int f\,d\mu \leq \int g\,d\mu .$

Monotonicity

: Suppose ${f k} k \in N$ is a sequence of non-negative measurable functions such that
$f_{k}(x)\leq f_{k+1}(x)\quad \forall k\in \mathbb {N} ,\,\forall x\in E.$ Then, the pointwise limit $f$ of $f k$ is Lebesgue measurable and $\lim _{k}\int f_{k}\,d\mu =\int f\,d\mu .$ The value of any of the integrals is allowed to be infinite.

Monotone convergence theorem

: If ${f k} k \in N$ is a sequence of non-negative measurable functions, then
$\int \liminf _{k}f_{k}\,d\mu \leq \liminf _{k}\int f_{k}\,d\mu .$ Again, the value of any of the integrals may be infinite.

Fatou's lemma

: Suppose ${f k} k \in N$ is a sequence of complex measurable functions with pointwise limit $f$ , and there is a Lebesgue integrable function $g$ (i.e., $g$ belongs to the $space L 1)$ such that $| f k | \leq g$ for all $k$ . Then $f$ is Lebesgue integrable and
$\lim _{k}\int f_{k}\,d\mu =\int f\,d\mu .$

Dominated convergence theorem

Two functions are said to be equal almost everywhere ( $f\ {\stackrel {\text{a.e.}}{=}}\ g$ for short) if $\{x\mid f(x)\neq g(x)\}$ is a subset of a null set. Measurability of the set $\{x\mid f(x)\neq g(x)\}$ is not required.

The following theorems are proved in most textbooks on measure theory and Lebesgue integration.^[7]

Necessary and sufficient conditions for the interchange of limits and integrals were proved by Cafiero,^[8]^[9]^[10]^[11] generalizing earlier work of Renato Caccioppoli, Vladimir Dubrovskii, and Gaetano Fichera.^[12]

for a non-technical description of Lebesgue integration

Henri Lebesgue

Null set

Integration

Measure

Sigma-algebra

Lebesgue space

Lebesgue–Stieltjes integration

Riemann integral

Henstock–Kurzweil integral

Bartle, Robert G. (1995). The elements of integration and Lebesgue measure. Wiley Classics Library. New York: John Wiley & Sons Inc. xii+179. 0-471-04222-6. MR 1312157.

ISBN

Bauer, Heinz (2001). Measure and Integration Theory. De Gruyter Studies in Mathematics 26. Berlin: De Gruyter. 236. 978-3-11-016719-1.

ISBN

(2004). Integration. I. Chapters 1–6. Translated from the 1959, 1965 and 1967 French originals by Sterling K. Berberian. Elements of Mathematics (Berlin). Berlin: Springer-Verlag. xvi+472. ISBN 3-540-41129-1. MR 2018901.

Bourbaki, Nicolas

Dudley, Richard M. (1989). Real analysis and probability. The Wadsworth & Brooks/Cole Mathematics Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software. xii+436. 0-534-10050-3. MR 0982264. Very thorough treatment, particularly for probabilists with good notes and historical references.

ISBN

Folland, Gerald B. (1999). Real analysis: Modern techniques and their applications. Pure and Applied Mathematics (New York) (Second ed.). New York: John Wiley & Sons Inc. xvi+386. 0-471-31716-0. MR 1681462.

ISBN

(1950). Measure Theory. New York, N. Y.: D. Van Nostrand Company, Inc. pp. xi+304. MR 0033869. A classic, though somewhat dated presentation.

Halmos, Paul R.

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Lebesgue integral"

(1904), Leçons sur l'intégration et la recherche des fonctions primitives, Paris: Gauthier-Villars

Lebesgue, Henri

(1972). Oeuvres scientifiques (en cinq volumes) (in French). Geneva: Institut de Mathématiques de l'Université de Genève. p. 405. MR 0389523.

Lebesgue, Henri

; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. Vol. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.

Lieb, Elliott

Loomis, Lynn H. (1953). An introduction to abstract harmonic analysis. Toronto-New York-London: D. Van Nostrand Company, Inc. pp. x+190. 0054173. Includes a presentation of the Daniell integral.

MR

(1974), Elementary classical analysis, W. H. Freeman.

Marsden

Munroe, M. E. (1953). Introduction to measure and integration. Cambridge, Mass.: Addison-Wesley Publishing Company Inc. pp. x+310. 0053186. Good treatment of the theory of outer measures.

MR

Royden, H. L. (1988). Real analysis (Third ed.). New York: Macmillan Publishing Company. pp. xx+444. 0-02-404151-3. MR 1013117.

ISBN

(1976). Principles of mathematical analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill Book Co. pp. x+342. MR 0385023. Known as Little Rudin, contains the basics of the Lebesgue theory, but does not treat material such as Fubini's theorem.

Rudin, Walter

Rudin, Walter (1966). Real and complex analysis. New York: McGraw-Hill Book Co. pp. xi+412. 0210528. Known as Big Rudin. A complete and careful presentation of the theory. Good presentation of the Riesz extension theorems. However, there is a minor flaw (in the first edition) in the proof of one of the extension theorems, the discovery of which constitutes exercise 21 of Chapter 2.

MR

(1937). Theory of the Integral. Monografie Matematyczne. Vol. 7 (2nd ed.). Warszawa-Lwów: G.E. Stechert & Co. JFM 63.0183.05. Zbl 0017.30004.. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach.

Saks, Stanisław

Shilov, G. E.; Gurevich, B. L. (1977). Integral, measure and derivative: a unified approach. Translated from the Russian and edited by Richard A. Silverman. Dover Books on Advanced Mathematics. New York: Dover Publications Inc. xiv+233. 0-486-63519-8. MR 0466463. Emphasizes the Daniell integral.

ISBN

Siegmund-Schultze, Reinhard (2008), "Henri Lebesgue", in Timothy Gowers; June Barrow-Green; Imre Leader (eds.), Princeton Companion to Mathematics, Princeton University Press.

. Topics in Real and Functional Analysis. (lecture notes).

Teschl, Gerald

(1957), Geometric Integration Theory, Princeton Mathematical Series, vol. 21, Princeton, NJ and London: Princeton University Press and Oxford University Press, pp. XV+387, MR 0087148, Zbl 0083.28204.

Whitney, H.

Yeh, James (2006). Real Analysis: Theory of Measure and Integral 2nd. Edition Paperback. Singapore: World Scientific Publishing Company Pte. Ltd. p. 760. 978-981-256-6.