Lebesgue integration

The integral of a positive function f between limits a and b can be interpreted as the area under the graph of f. This is straightforward for functions such as polynomials, but what does it mean for more exotic functions? In general, for which class of functions does "area under the curve" make sense? The answer to this question has great theoretical and practical importance.

As part of a general movement toward rigor in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on a firm foundation. The Riemann integral—proposed by Bernhard Riemann (1826–1866)—is a broadly successful attempt to provide such a foundation. Riemann's definition starts with the construction of a sequence of easily calculated areas that converge to the integral of a given function. This definition is successful in the sense that it gives the expected answer for many already-solved problems, and gives useful results for many other problems.

However, Riemann integration does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyze. This is important, for instance, in the study of Fourier series, Fourier transforms, and other topics. The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign (via the monotone convergence theorem and dominated convergence theorem).

While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition considers horizontal slabs that are not necessarily just rectangles, and so it is more flexible. For this reason, the Lebesgue definition makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but does not have a Riemann integral. Furthermore, the Lebesgue integral of this function is zero, which agrees with the intuition that when picking a real number uniformly at random from the unit interval, the probability of picking a rational number should be zero.

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. 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.

— Source: (Siegmund-Schultze 2008)

The insight is that one should be able to rearrange the values of a function freely, while preserving the value of the integral. This process of rearrangement can convert a very pathological function into one that is "nice" from the point of view of integration, and thus let such pathological functions be integrated.

Intuitive interpretation

A measurable function is shown, together with the set ${\displaystyle \{x:f(x)>t\}}$ (on the x-axis). The Lebesgue integral is obtained by slicing along the y-axis, using the 1-dimensional Lebesgue measure to measure the "width" of the slices.

Folland (1984) 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."[1]

For the Riemann integral, the domain is partitioned into intervals, and bars are constructed to meet the height of the graph. The areas of these bars are added together, and this approximates the integral, in effect by summing areas of the form ${\displaystyle f(x)dx}$ where ${\displaystyle f(x)}$ is the height of a rectangle and ${\displaystyle dx}$ is its width.

For the Lebesgue integral, the range is partitioned into intervals, and so the region under the graph is partitioned into horizontal "slabs" (which may not be connected sets). The area of a small horizontal "slab" under the graph of f, of height dy, is equal to the measure of the slab's width times dy:

${\displaystyle \mu \left(\{x\mid f(x)>y\}\right)\,dy.}$

The Lebesgue integral may then be defined by adding up the areas of these horizontal slabs.

Simple functions

Riemannian (top) vs Lebesgue (bottom) integration of smoothed COVID-19 daily case data from Serbia (Summer-Fall 2021).

An equivalent way to introduce the Lebesgue integral is to use so-called simple functions, which generalize the step functions of Riemann integration. Consider, for example, determining the cumulative COVID-19 case count from a graph of smoothed new daily cases (right).

The Riemann–Darboux approach

Partition the domain (time period) into intervals (eight, in the example at right) and construct bars with heights that meet the graph. The cumulative count is found by summing, over all bars, the product of interval width (time in days) and the bar height (cases per day).

The Lebesgue approach

Choose a finite number of target values (eight, in the example) in the range of the function. By constructing bars with heights equal to these values, but below the function, they imply a partitioning of the domain into the same number of subsets (subsets, indicated by color in the example, need not be connected). This is a "simple function," as described below. The cumulative count is found by summing, over all subsets of the domain, the product of the measure on that subset (total time in days) and the bar height (cases per day).

The theory of the Lebesgue integral requires a theory of measurable sets and measures on these sets, as well as a theory of measurable functions and integrals on these functions.

Via simple functions

Approximating a function by a simple function.

One approach to constructing the Lebesgue integral is to make use of so-called simple functions: finite, real linear combinations of indicator functions. Simple functions that lie directly underneath a given function f can be constructed by partitioning the range of f into a finite number of layers. The intersection of the graph of f with a layer identifies a set of intervals in the domain of f, which, taken together, is defined to be the preimage of the lower bound of that layer, under the simple function. In this way, the partitioning of the range of f implies a partitioning of its domain. 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. The integral of a non-negative general measurable function is then defined as an appropriate supremum of approximations by simple functions, and the integral of a (not necessarily positive) measurable function is the difference of two integrals of non-negative measurable functions.

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

• ${\displaystyle 1_{\mathbf {Q} }}$ 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 Darboux sums are all one, and the lower Darboux sums are all zero.
• ${\displaystyle 1_{\mathbf {Q} }}$ is Lebesgue-integrable on [0, 1] using the Lebesgue measure: Indeed, it is the indicator function of the rationals so by definition
  $${\displaystyle \int _{[0,1]}1_{\mathbf {Q} }\,d\mu =\mu (\mathbf {Q} \cap [0,1])=0,}$$

  
  because Q is countable.

A technical issue in Lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation. In elementary calculus, one defines integration with respect to an orientation:

${\displaystyle \int _{b}^{a}f:=-\int _{a}^{b}f.}$

Generalizing this to higher dimensions yields integration of differential forms. By contrast, Lebesgue integration provides an alternative generalization, integrating over subsets with respect to a measure; this can be notated as

${\displaystyle \int _{A}f\,d\mu =\int _{[a,b]}f\,d\mu }$

to indicate integration over a subset A. For details on the relation between these generalizations, see Differential form § Relation with measures.

With the advent of Fourier series, many analytical problems involving integrals came up whose satisfactory solution required interchanging limit processes and integral signs. However, the conditions under which the integrals

${\displaystyle \sum _{k}\int f_{k}(x)dx,\quad \int \left[\sum _{k}f_{k}(x)\right]dx}$

are equal proved quite elusive in the Riemann framework. There are some other technical difficulties with the Riemann integral. These are linked with the limit-taking difficulty discussed above.

Failure of monotone convergence. As shown above, the indicator function 1_Q on the rationals is not Riemann integrable. In particular, the Monotone convergence theorem fails. To see why, let {a_k} be an enumeration of all the rational numbers in [0, 1] (they are countable so this can be done.) Then let

${\displaystyle g_{k}(x)={\begin{cases}1&{\text{if }}x=a_{j},j\leq k\\0&{\text{otherwise}}\end{cases}}}$

The function g_k is zero everywhere, except on a finite set of points. Hence its Riemann integral is zero. Each g_k is non-negative, and this sequence of functions is monotonically increasing, but its limit as k → ∞ is 1_Q, which is not Riemann integrable.

Unsuitability for unbounded intervals. The Riemann integral can only integrate functions on a bounded interval. It can however be extended to unbounded intervals by taking limits, so long as this doesn't yield an answer such as ∞ − ∞.

Integrating on structures other than Euclidean space. The Riemann integral is inextricably linked to the order structure of the real line.

Two functions are said to be equal almost everywhere (${\displaystyle f\ {\stackrel {\text{a.e.}}{=}}\ g}$ for short) if ${\displaystyle \{x\mid f(x)\neq g(x)\}}$ is a subset of a null set.

Measurability of the set ${\displaystyle \{x\mid f(x)\neq g(x)\}}$ is not required.

• If f, g are non-negative measurable functions (possibly assuming the value +∞) such that f = g almost everywhere, then
  $${\displaystyle \int f\,d\mu =\int g\,d\mu .}$$

  
  To wit, the integral respects the equivalence relation of almost-everywhere equality.
• If f, 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.
• Linearity: If f and g are Lebesgue integrable functions and a and b are real numbers, then af + bg is Lebesgue integrable and
  $${\displaystyle \int (af+bg)\,d\mu =a\int f\,d\mu +b\int g\,d\mu .}$$

  
• Monotonicity: If f ≤ g, then
  $${\displaystyle \int f\,d\mu \leq \int g\,d\mu .}$$

  
• Let ${\displaystyle (\Omega ,\Sigma ,\mu )}$ be a measure space. Denote ${\displaystyle \operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}}}$ the ${\displaystyle \sigma }$-algebra of Borel sets on ${\displaystyle [0,+\infty ]}$. (By definition, ${\displaystyle \operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}}}$ contains the set ${\displaystyle \{+\infty \}}$ and all Borel subsets of ${\displaystyle \mathbb {R} _{\geq 0}}$.) Consider a ${\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})}$-measurable non-negative function ${\displaystyle s:\Omega \to [0,+\infty ]}$. For a set ${\displaystyle S\in \Sigma }$, define
  $${\displaystyle \nu (S)=\int _{S}s\,d\mu .}$$

  
  Then ${\displaystyle \nu }$ is a Lebesgue measure on ${\displaystyle (\Omega ,\Sigma )}$.
• Monotone convergence theorem: Suppose { f_k}_{k ∈ N} is a sequence of non-negative measurable functions such that
  $${\displaystyle f_{k}(x)\leq f_{k+1}(x)\quad \forall k\in \mathbb {N} ,\,\forall x\in E.}$$

  
  Then, the pointwise limit f of f_k is Lebesgue measurable and
  $${\displaystyle \lim _{k}\int f_{k}\,d\mu =\int f\,d\mu .}$$

  
  The value of any of the integrals is allowed to be infinite.
• Fatou's lemma: If { f_k}_{k ∈ N} is a sequence of non-negative measurable functions, then
  $${\displaystyle \int \liminf _{k}f_{k}\,d\mu \leq \liminf _{k}\int f_{k}\,d\mu .}$$

  
  Again, the value of any of the integrals may be infinite.
• Dominated convergence theorem: Suppose {f_k}_{k ∈ 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 L¹) such that | f_k | ≤ g for all k.
  Then, f is Lebesgue integrable and
  $${\displaystyle \lim _{k}\int f_{k}\,d\mu =\int f\,d\mu .}$$

  

It is possible to develop the integral with respect to the Lebesgue measure without relying on the full machinery of measure theory. One such approach is provided by the Daniell integral.

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). Integrals of more general functions can be built starting from these integrals.

Let C_c be the space of all real-valued compactly supported continuous functions of R. Define a norm on C_c by

$${\displaystyle \left\|f\right\|=\int |f(x)|\,dx.}$$

Then C_c is a normed vector space (and in particular, it is a metric space.) All metric spaces have Hausdorff completions, so let L¹ be its completion. This space is isomorphic to the space of Lebesgue integrable functions modulo the subspace of functions with integral zero. Furthermore, the Riemann integral ∫ is a uniformly continuous functional with respect to the norm on C_c, which is dense in L¹. Hence ∫ has a unique extension to all of L¹. This integral is precisely the Lebesgue integral.

More generally, when the measure space on which the functions are defined is also a locally compact topological space (as is the case with the real numbers R), measures compatible with the topology in a suitable sense (Radon measures, of which the Lebesgue measure is an example) an integral with respect to them can be defined in the same manner, starting from the integrals of continuous functions with compact support. 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. The value of a measure at a compactly supported function is then also by definition the integral of the function. One then proceeds to expand the measure (the integral) to more general functions by continuity, and defines the measure of a set as the integral of its indicator function. This is the approach taken by Bourbaki (2004) and a certain number of other authors. For details see Radon measures.

The main purpose of the Lebesgue integral is to provide an integral notion where limits of integrals hold under mild assumptions. There is no guarantee that every function is Lebesgue integrable. But it may happen that improper integrals exist for functions that are not Lebesgue integrable. One example would be the sinc function:

$${\displaystyle {\frac {\sin(x)}{x}}}$$

over the entire real line. This function is not Lebesgue integrable, as

$${\displaystyle \int _{-\infty }^{\infty }\left|{\frac {\sin(x)}{x}}\right|dx=\infty .}$$

On the other hand, ${\textstyle \int _{-\infty }^{\infty }{\frac {\sin(x)}{x}}dx}$ exists as an improper integral and can be computed to be finite; it is twice the Dirichlet integral.

• Henri Lebesgue, for a non-technical description of Lebesgue integration
• Null set
• Integration
• Measure
• Sigma-algebra
• Lebesgue space
• Lebesgue–Stieltjes integration
• Riemann integral
• Henstock–Kurzweil integral

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