Integral

One of the central notions in mathematical analysis and all of mathematics, which arose in connection with two problems: to recover a function from its derivative (for example, the problem of finding the law of motion of a material object along a straight line when the velocity of this point is known); and to calculate the area bounded by the graph of a function $f$ on an interval $a\leq x\leq b$ and the $x$-axis (the problem of calculating the work performed by a force over an interval of time $a\leq t\leq b$ leads to this problem, as do other problems).

The two problems indicated above lead to two forms of the integral, the indefinite and the definite integral. The study of the properties and calculation of these interrelated forms of the integral constitutes the problem of integral calculus.

In the course of development of mathematics and under the influence of the requirements of natural science and technology, the notions of the indefinite and the definite integral have undergone a number of generalizations and modifications.

The indefinite integral.

A primitive of a function $f$ of the variable $x$ on an interval $a<x<b$ is any function $F$ whose derivative is equal to $f$ at each point $x$ of the interval. It is clear that if $F$ is a primitive of $f$ on the interval $a<x<b$, then so is $F_1=F+C$, where $C$ is an arbitrary constant. The converse also holds: Any two primitives of the same function $f$ on the interval $a<x<b$ can only differ by a constant. Consequently, if $F$ is one of the primitives of $f$ on the interval $a<x<b$, then any primitive of $f$ on this interval has the form $F+C$, where $C$ is a constant. The collection of all primitives of $f$ on the interval $a<x<b$ is called the indefinite integral of $f$ (on this interval) and is denoted by the symbol

$$\int f(x)\,dx.$$

According to the fundamental theorem of integral calculus, there exists for each continuous function $f$ on the interval $a<x<b$ a primitive, and hence an indefinite integral, on this interval (cf. also Indefinite integral).

The definite integral.

The notion of the definite integral is introduced either as a limit of integral sums (see Cauchy integral; Riemann integral; Lebesgue integral; Stieltjes integral) or, in the case when the given function $f$ is defined on some interval $[a,b]$ and has a primitive $F$ on this interval, as the difference between the values at the end points, that is, as $F(b)-F(a)$. The definite integral of $f$ on $[a,b]$ is denoted by $\int_a^bf(x)\,dx$. The definition of the integral as a limit of integral sums for the case of continuous functions was stated by A.L. Cauchy in 1823. The case of arbitrary functions was studied by B. Riemann (1853). A substantial advance in the theory of definite integrals was made by G. Darboux (1879), who introduced the notion of upper and lower Riemann sums (see Darboux sum). A necessary and sufficient condition for the Riemann integrability of discontinuous functions was established in final form in 1902 by H. Lebesgue.

There is the following relationship between the definitions of the definite integral of a continuous function $f$ on a closed interval $[a,b]$ and the indefinite integral (or primitive) of this function: 1) if $F$ is any primitive of $f$, then the following Newton–Leibniz formula holds:

$$\int\limits_a^bf(x)\,dx=F(b)-F(a);$$

2) for any $x$ in the interval $[a,b]$, the indefinite integral of the continuous function $f$ can be written in the form

$$\int f(x)\,dx=\int\limits_a^xf(t)\,dt+C,$$

where $C$ is an arbitrary constant. In particular, the definite integral with variable upper limit,

$$F(x)=\int\limits_a^xf(t)\,dt\label{1}\tag{1}$$

is a primitive of $f$.

In order to introduce the definite integral of $f$ over $[a,b]$ in the sense of Lebesgue, the set of values of $y$ is divided into subintervals of points $\ldots<y_{-1}<y_0<y_1<\dots$, and one denotes by $M_i$ the set of all values of $x$ in the interval $[a,b]$ for which $y_{i-1}\leq f(x)<y_i$, and by $\mu(M_i)$ the measure of the set $M_i$ in the sense of Lebesgue (cf. Lebesgue measure). A Lebesgue integral sum of the function $f$ on the interval $[a,b]$ is defined by the formula

$$\sigma=\sum_i\eta_i\mu(M_i),\label{2}\tag{2}$$

where $\eta_i$ are arbitrary numbers in the interval $[y_{i-1},y_i]$.

A function $f$ is said to be Lebesgue integrable on the interval $[a,b]$ if the limit of the integral sums \eqref{2} exists and is finite as the maximum width of the intervals $(y_{i-1},y_i)$ tends to zero, that is, if there exists a real number $I$ such that for any $\epsilon>0$ there is a $\delta>0$ such that under the single condition $\max(y_i-y_{i-1})<\delta$ the inequality $|\sigma-I|<\epsilon$ holds. The limit $I$ is then called the definite Lebesgue integral of $f$ over $[a,b]$.

Instead of the interval $[a,b]$ one can consider an arbitrary set that is measurable with respect to some non-negative complete countably-additive measure. An alternative introduction to the Lebesgue integral can be given, when one defines this integral originally on the set of so-called simple functions (that is, measurable functions assuming at most a countable number of values), and then introduces the integral by means of a limit transition for any function that can be expressed as the limit of a uniformly-convergent sequence of simple functions (see Lebesgue integral).

Each Riemann-integrable function is Lebesgue integrable. The converse is false, since there exist Lebesgue-integrable functions that are discontinuous on a set of positive measure (for example, the Dirichlet function).

In order that a bounded function be Lebesgue integrable, it is necessary and sufficient that this function belongs to the class of measurable functions (cf. Measurable function). The functions encountered in mathematical analysis are, as a rule, measurable. This means that the Lebesgue integral has a generality that is sufficient for the requirements of analysis.

The Lebesgue integral also covers the cases of absolutely-convergent improper integrals (cf. Improper integral).

The generality attained by the definition of the Lebesgue integral is absolutely essential in many questions in modern mathematical analysis (the theory of generalized functions, the definition of generalized solutions of differential equations, and the isomorphism of the Hilbert spaces $L_2$ and $l_2$, which is equivalent to the so-called Riesz–Fischer theorem in the theory of trigonometric or arbitrary orthogonal series; all these theories have proved possible only by taking the integral to be in the sense of Lebesgue).

The primitive in the sense of Lebesgue is naturally defined by means of equation \eqref{1}, in which the integral is taken in the sense of Lebesgue. The relation $F'=f$ in this case holds everywhere, except perhaps on a set of measure zero.

Other generalizations of the notions of an integral.

In 1894 T.J. Stieltjes gave another generalization of the Riemann integral (which acquired the name of Stieltjes integral), important for applications, in which one considers the integrability of a function $f$ defined on some interval $[a,b]$ with respect to a second function defined on the same interval. The Stieltjes integral of $f$ with respect to the function $U$ is denoted by the symbol

$$I=\int\limits_a^bf(x)\,dU(x).\label{3}\tag{3}$$

If $U$ has a bounded Riemann-integrable derivative $U'$, then the Stieltjes integral reduces to the Riemann integral by the formula

$$\int\limits_a^bf(x)\,dU(x)=\int\limits_a^bf(x)U'(x)\,dx.$$

In particular, when $U(x)=x+C$, the Stieltjes integral \eqref{3} is the Riemann integral $\int_a^bf(x)\,dx$.

However, the interesting case for applications is when the function $U$ does not have a derivative. An example of such a $U$ is the spectral measure in the study of spectral decompositions.

The curvilinear integral

$$\int\limits_\Gamma f(x,y)\,dx$$

along the curve $\Gamma$ defined by the equations $x=\phi(t),y=\psi(t)$, $a\leq t\leq b$, is a special case of the Stieltjes integral, since it can be written in the form

$$\int\limits_a^bf[\phi(t),\psi(t)]\,d\phi(t).$$

A further generalization of the notion of the integral is obtained by integration over an arbitrary set in a space of any number of variables. In the most general case it is convenient to regard the integral as a function of the set $M$ over which the integration is carried out (see Set function), in the form

$$F(M)=\int\limits_Mf(x)\,dU(x),$$

where $U$ is a set function on $M$ (its measure in a particular case) and the points belong to the set $M$ over which the integration proceeds. Particular cases of this type of integration are multiple integrals and surface integrals (cf. Multiple integral; Surface integral).

Another generalization of the notion of the integral is that of the improper integral.

In 1912 A. Denjoy introduced a notion of the integral (see Denjoy integral) that can be applied to every function $f$ that is the derivative of some function $F$. This enables one to reduce the constructive definition of the integral to a degree of generality which completely answers the problem of finding a definite integral taken in the sense of a primitive.

References

[1]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1971–1973) (Translated from Russian)
[2]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801
[3]	L.D. Kudryavtsev, "Mathematical analysis" , Moscow (1973) (In Russian) MR1617334 MR1070567 MR1070566 MR1070565 MR0866891 MR0767983 MR0767982 MR0628614 MR0619214 Zbl 1080.00002 Zbl 1080.00001 Zbl 1060.26002 Zbl 0869.00003 Zbl 0696.26002 Zbl 0703.26001 Zbl 0609.00001 Zbl 0632.26001 Zbl 0485.26002 Zbl 0485.26001
[4]	S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1975) (Translated from Russian) Zbl 1029.00003 Zbl 0727.00001 Zbl 0723.00003 Zbl 0397.00003 Zbl 0384.00004
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[6]	H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928) MR2857993 Zbl 54.0257.01

Comments

Concerning the "simple functions" mentioned above: every real-valued measurable function is the limit of a uniformly-convergent sequence of simple functions. However, such functions need not be Lebesgue integrable.

There are many other types of integrals besides those of Riemann and Lebesgue, cf., e.g., $A$-integral; Boks integral; Burkill integral; Daniell integral; Darboux sum; Kolmogorov integral; Perron integral; Pettis integral; Radon integral; Repeated integral; Strong integral; Wiener integral.

References

[a1]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
[a2]	E.J. MacShane, "Integration" , Princeton Univ. Press (1944) Zbl 1088.26500
[a3]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24 MR0344043 Zbl 0278.26001
[a4]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05
[a5]	K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) MR0604364 Zbl 0454.26001
[a6]	A.J. Weir, "Lebesgue integration and measure" , Cambridge Univ. Press (1985) MR0480918 MR0480919 Zbl 0257.26001
[a7]	A.C. Zaanen, "Integration" , North-Holland (1967) MR0222234 Zbl 0175.05002
[a8]	G.E. Shilov, B.L. Gurevich, "Integral, measure, and derivative: a unified approach" , Prentice-Hall (1966) (Translated from Russian) MR0194571 Zbl 0138.27501
[a9]	I.N. Pesin, "Classical and modern integration theories" , Acad. Press (1970) (Translated from Russian) MR0264015 Zbl 0206.06401
[a10]	J. Diestel, J.J. Uhl jr., "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977) MR0453964 Zbl 0369.46039