Lebesgue Integral

The Lebesgue integral is a definition of the integral that uses the measures of sets to aggregate the values of a function. On Euclidean space, it ordinarily uses Lebesgue measure, which generalizes length, area, and volume; the same construction also works with other measures. It integrates a broader class of functions than the Riemann integral and provides precise conditions for exchanging integration with limiting operations. It is a fundamental construction in measure theory. (math.ucdavis.edu)

Historical background and guiding idea

Henri Lebesgue developed the theory at the beginning of the twentieth century. His thesis, Intégrale, longueur, aire, appeared in 1902 and treated integration alongside questions concerning length and area. Lebesgue himself identified this publication in a subsequent paper on the problem of areas. (numdam.org)

A Riemann sum samples function values on small intervals of the domain and multiplies them by interval lengths. Lebesgue integration instead permits approximation by functions that are constant on measurable sets, which need not be intervals. Informally, it groups points according to their function values and measures how large those groups are. This description is a useful intuition, rather than the formal definition. (arxiv.org)

Measure spaces and formal construction

The construction starts with a measure space $(X,\Sigma,\mu)$. Here $X$ is a set, $\Sigma$ is a sigma-algebra of measurable subsets, and $\mu$ is a nonnegative, countably additive measure. A measurable function has measurable inverse images of Borel sets; for real-valued functions, it suffices that $\{x:f(x)>a\}$ be measurable for every real $a$. (math.ucdavis.edu)

A nonnegative simple function takes finitely many values and can be written

where the $A_k$ are disjoint measurable sets, $a_k\geq0$, and $\mathbf 1_{A_k}$ is their indicator function. Define

with $0\cdot\infty=0$. For a nonnegative measurable function $f$, define

The result may be infinite. (math.ucdavis.edu)

For a signed function, set $f^+=\max(f,0)$ and $f^-=\max(-f,0)$. Its integral is

whenever this does not require subtracting infinity from infinity. The function is Lebesgue integrable when $\int_X|f|\,d\mu<\infty$. Functions taking complex values are integrated through their real and imaginary parts. (logic.ucla.edu)

Basic properties and comparison with Riemann integration

For integrable functions, integration is linear, preserves inequalities, and satisfies

Two measurable functions equal almost everywhere have the same integral: changing values on a set of measure zero does not affect it. Thus integrals depend on equivalence classes rather than on every individual pointwise value. (logic.ucla.edu)

On a compact interval, every Riemann-integrable function is Lebesgue integrable, and the two integrals agree. Lebesgue’s criterion states that a bounded function is Riemann integrable exactly when its discontinuities form a set of Lebesgue measure zero. In particular, every continuous function on such an interval is integrable in both senses. (maths-people.anu.edu.au)

For example, let $f$ equal one at rational points of $[0,1]$ and zero elsewhere. The rationals form a countable set of measure zero, so its Lebesgue integral is zero. Nevertheless, every interval contains rational and irrational points, making its lower Riemann sums zero and its upper sums one. It is therefore not Riemann integrable. (arxiv.org)

An improper Riemann integral that converges only through cancellation need not define a Lebesgue-integrable function. Finite Lebesgue integrability requires absolute integrability, not merely convergence of a particular sequence of truncated integrals. (maths-people.anu.edu.au)

Convergence theorems

Three results describe the interaction between integration and limits:

• The monotone convergence theorem states that if $0\leq f_n\uparrow f$ almost everywhere, then

  $$\int_X f\,d\mu=\lim_{n\to\infty}\int_X f_n\,d\mu,$$

  allowing infinite values.

• Fatou’s lemma states that for nonnegative measurable functions,

  $$\int_X\liminf_{n\to\infty}f_n\,d\mu \leq\liminf_{n\to\infty}\int_Xf_n\,d\mu.$$

• The dominated convergence theorem states that if $f_n\to f$ almost everywhere and $|f_n|\leq g$ almost everywhere for a single integrable function $g$, then $f$ is integrable and

  $$\int_X|f_n-f|\,d\mu\to0.$$

  Consequently, their integrals converge. (arxiv.org)

Pointwise convergence alone is insufficient. For example, $f_n=n\mathbf1_{(0,1/n)}$ on $[0,1]$ converges everywhere to zero, while every integral equals one. The domination hypothesis prevents this concentration of mass from invalidating passage to the limit. (maths-people.anu.edu.au)

Product integration and applications

For sigma-finite measure spaces, Tonelli’s theorem allows a nonnegative measurable function on a product space to be integrated in either order, even when the result is infinite. Fubini’s theorem gives the corresponding equality for absolutely integrable functions. These results supply the conditions needed to justify iterated integrals. (math.ucdavis.edu)

In probability, a probability space is a measure space of total measure one. The expected value of an integrable random variable $Y$ is

This formulation accommodates discrete, continuous, and mixed distributions within one definition. (logic.ucla.edu)

In functional analysis, Lebesgue integration defines the spaces $L^p$, whose elements identify functions equal almost everywhere. For $1\leq p<\infty$,

These are complete normed spaces, or Banach spaces. The space $L^2$, with inner product $\int_Xf\overline g\,d\mu$, is a Hilbert space. (maths-people.anu.edu.au)