Product Integral

A product integral is any Product (mathematics), product-based counterpart of the usual Summation, sum-based integral of calculus. The first product integral (''#Type I: Volterra integral, Type I'' below) was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations.
A. Slavík
''Product integration, its history and applications''
, Matfyzpress, Prague, 2007.
Other examples of product integrals are the #Type II: geometric integral, geometric integral (''#Type II: geometric integral, Type II'' below), the #Type III: bigeometric integral, bigeometric integral (''#Type III: bigeometric integral, Type III'' below), and some other integrals of non-Newtonian calculus.
Michael Grossman
''The First Nonlinear System of Differential And Integral Calculus''
, 1979.Michael Grossman
''Bigeometric Calculus: A System with a Scale-Free Derivative''
, 1983.

Product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic population dynamics using multiplication integrals (multigrals), analysis (mathematics), analysis and quantum mechanics. The #Type II: geometric integral, geometric integral, together with the #Results, geometric derivative, is useful in image analysis
Luc Florack and Hans van Asse
"Multiplicative calculus in biomedical image analysis"
Journal of Mathematical Imaging and Vision, , 2011.
and in the study of growth/decay phenomena (e.g., in economic growth, bacterial growth, and radioactive decay).
Agamirza E. Bashirov, Emine Misirli, Yucel Tandogdu, and Ali Ozyapic
"On modelling with multiplicative differential equations"
Applied Mathematics – A Journal of Chinese Universities, Volume 26, Number 4, pages 425–428, , Springer, 2011.
The #Type III: bigeometric integral, bigeometric integral, together with the bigeometric derivative, is useful in some applications of fractals,Marek Rybaczuk, Alicja Kedzia and Witold Zielinski (2001
"The concept of physical and fractal dimension II. The differential calculus in dimensional spaces"
''Chaos, Solitons, & Fractals''Volume 12, Issue 13, October 2001, pages 2537–2552.

and in the theory of Elasticity of a function, elasticity in economics.

This article adopts the "product" $\prod$ notation for product integration instead of the "integral" $\int$ (usually modified by a superimposed "times" symbol or letter P) favoured by Vito Volterra, Volterra and others. An arbitrary classification of types is also adopted to impose some order in the field.

Basic definitions

The classical Riemann integral of a Function (mathematics), function $f:[a,b]\to\mathbb$ can be defined by the relation

:$\int_a^b f(x)\,dx = \lim_\sum f(x_i)\,\Delta x,$

where the Limit (mathematics), limit is taken over all Partition of an interval, partitions of the Interval (mathematics), interval $[a,b]$ whose Partition of an interval#Norm of a partition, norms approach zero.

Roughly speaking, product integrals are similar, but take the Limit (mathematics), limit of a Product (mathematics), product instead of the Limit (mathematics), limit of a Summation, sum. They can be thought of as "Mathematical analysis, continuous" versions of "Discrete mathematics, discrete" Product (mathematics), products.

The most popular product integrals are the following:

Type I: Volterra integral

:$\prod_a^b \big(1 + f(x)\,dx\big) = \lim_ \prod \big(1 + f(x_i)\,\Delta x\big).$

The type I product integral corresponds to Vito Volterra, Volterra's original definition. The following relationship exists for scalar functions $f:[a,b] \to \mathbb$:

:$\prod_a^b \big(1 + f(x)\,dx\big) = \exp\left(\int_a^b f(x) \,dx\right),$

which is not a Multiplicative function, multiplicative Functional (mathematics), operator. (So the concepts of product integral and Multiplicative function, multiplicative integral are not the same).

The Volterra product integral is most useful when applied to matrix-valued functions or functions with values in a Banach algebra, where the last equality is no longer true (see the references below).

When applied to scalars belonging to a non-commutative field, to matrixes, and to operators, i.e. to mathematical objects that don't commute, the Volterra integral splits in two definitions

Left Product integral
:$P(A,D)=\prod_^(\mathbb+A(\xi_i)\Delta t_i) = (\mathbb+A(\xi_m)\Delta t_m) \cdots (\mathbb+A(\xi_1)\Delta t_1)$
With the notation of left products (i.e. normal products applied from left)
:$\prod_a^b (\mathbb+A(t)dt)=\lim_ P(A,D)$

Right Product Integral
:$P(A,D)^*=\prod_^(\mathbb+A(\xi_i)\Delta t_i) = (\mathbb+A(\xi_1)\Delta t_1) \cdots (\mathbb+A(\xi_m)\Delta t_m)$
With the notation of right products (i.e. applied from right)
:$(\mathbb+A(t)dt) \prod_a^b =\lim_ P(A,D)^*$

Where $\mathbb$ is the identity matrix and D is a partition of the interval [a,b] in the Riemann sense, i.e. the limit is over the maximum interval in the partition.
Note how in this case time ordering comes evident in the definitions.

For Scalar (mathematics), scalar functions, the derivative in the Volterra system is the logarithmic derivative, and so the Volterra system is not a multiplicative calculus and is not a non-Newtonian calculus.

Type II: geometric integral

:$\prod_a^b f(x)^ = \lim_ \prod = \exp\left(\int_a^b \ln f(x) \,dx\right),$
which is called the geometric integral and is a Multiplicative function, multiplicative Functional (mathematics), operator.

This definition of the product integral is the Mathematical analysis, continuous analog of the Discrete mathematics, discrete Product (mathematics), product Functional (mathematics), operator
:$\prod_^b$
(with $i, a, b \in \mathbb$) and the Multiplicative function, multiplicative analog to the (normal/standard/Additive map, additive) integral
:$\int_a^b dx$
(with $x \in [a,b]$):

:

It is very useful in Probability theory, stochastics, where the Likelihood function#Log-likelihood, log-likelihood (i.e. the logarithm of a product integral of Independence (probability theory), independent random variables) equals the integral of the logarithm of these (infinitesimally many) random variables:

:$\ln \prod_a^b p(x)^ = \int_a^b \ln p(x) \,dx.$

Type III: bigeometric integral

:$\prod_a^b f(x)^ = \exp\left(\int_r^s \ln f(e^x) \,dx\right),$
where ''r'' = ln ''a'', and ''s'' = ln ''b''.

The type III product integral is called the bigeometric integral and is a Multiplicative function, multiplicative Functional (mathematics), operator.

Results

;Basic results

The following results are for the #Type II: geometric integral, type II product integral (the geometric integral). Other types produce other results.

: $\prod_a^b c^ = c^,$

: $\prod_a^b x^ = \frac ^,$

: $\prod_0^b x^ = b^b ^,$

: $\prod_a^b \left(f(x)^k\right)^ = \left(\prod_a^b f(x)^\right)^k,$

: $\prod_a^b \left(c^\right)^ = c^,$

The #Type II: geometric integral, geometric integral (type II above) plays a central role in the geometric calculus,
M. Grossman, R. Katz
''Non-Newtonian Calculus''
, Lee Press, 1972.
which is a multiplicative calculus. The inverse of the geometric integral, which is the geometric derivative, denoted $f^*(x)$, is defined using the following relationship:

: $f^*(x)=\exp\left(\frac\right)$
Thus, the following can be concluded:
;Fundamental theorem of calculus, The fundamental theorem

: $\prod_a^b f^*(x)^ = \prod_a^b \exp\left(\frac \,dx\right) = \frac,$

;Product rule

: $(fg)^* = f^* g^*.$

;Quotient rule

: $(f/g)^* = f^*/g^*.$

;Law of large numbers

: $\sqrt[n] \underset \prod_x X^,$

where X is a random variable with Probability distribution#Cumulative distribution function, probability distribution ''F''(''x'').

Compare with the standard law of large numbers:

: $\frac \underset \int X \,dF(x).$

Lebesgue-type product-integrals

Just like the Lebesgue integration, Lebesgue version of (classical) integrals, one can compute product integrals by approximating them with the product integrals of simple functions. Each type of product integral has a different form for simple functions.

Type I: Volterra integral

Because simple functions generalize step functions, in what follows we will only consider the special case of simple functions that are step functions. This will also make it easier to compare the Lebesgue integration, Lebesgue definition with the Riemann integral, Riemann definition.

Given a step function $f: [a,b] \to \mathbb$ with corresponding Partition of an interval, partition $a = y_0 < y_1 < \dots < y_m$ and a Partition of an interval#Tagged partitions, tagged partition

: $a = x_0 < x_1 < \dots < x_n = b, \quad x_0 \le t_0 \le x_1, x_1 \le t_1 \le x_2, \dots, x_ \le t_ \le x_n,$

one Approximation#Mathematics, approximation of the "Riemann definition" of the #Type I: Volterra integral, type I product integral is given byA. Slavík
''Product integration, its history and applications''
p. 65. Matfyzpress, Prague, 2007. .

: $\prod_^ \left[ \big(1 + f(t_k)\big) \cdot (x_ - x_k) \right].$

The (type I) product integral was defined to be, roughly speaking, the Limit (mathematics), limit of these Product (mathematics), products by Ludwig Schlesinger in a 1931 article.

Another approximation of the "Riemann definition" of the type I product integral is defined as

: $\prod_^ \exp\big(f(t_k) \cdot (x_ - x_k)\big).$

When $f$ is a constant function, the limit of the first type of approximation is equal to the second type of approximation.A. Slavík
''Product integration, its history and applications''
p. 71. Matfyzpress, Prague, 2007. . Notice that in general, for a step function, the value of the second type of approximation doesn't depend on the partition, as long as the partition is a Partition of an interval#Refinement of a partition, refinement of the partition defining the step function, whereas the value of the first type of approximation ''does'' depend on the Partition of an interval#Refinement of a partition, fineness of the partition, even when it is a refinement of the partition defining the step function.

It turns out thatA. Slavík
''Product integration, its history and applications''
p. 72. Matfyzpress, Prague, 2007. . that for ''any'' product-integrable function $f$, the limit of the first type of approximation equals the limit of the second type of approximation. Since, for step functions, the value of the second type of approximation doesn't depend on the fineness of the partition for partitions "fine enough", it makes sense to defineA. Slavík
''Product integration, its history and applications''
p. 80. Matfyzpress, Prague, 2007. the "Lebesgue (type I) product integral" of a step function as

: $\prod_a^b \big(1 + f(x) \,dx\big) \overset \prod_^ \exp\big(f(s_k) \cdot (y_ - y_k)\big),$

where $y_0 < a = s_0 < y_1 < \dots < y_ < s_ < y_n = b$ is a tagged partition, and again $a = y_0 < y_1 < \dots < y_m$ is the partition corresponding to the step function $f$. (In contrast, the corresponding quantity would not be unambiguously defined using the first type of approximation.)

This generalizes to List of mathematical jargon#Descriptive informalities, arbitrary measure spaces readily. If $X$ is a measure space with Measure (mathematics), measure $\mu$, then for any product-integrable simple function $f(x) = \sum_^n a_k I_(x)$ (i.e. a conical combination of the indicator functions for some Disjoint sets, disjoint Measure (mathematics)#Definition, measurable sets $A_0, A_1, \dots, A_ \subseteq X$), its type I product integral is defined to be

: $\prod_X \big(1 + f(x) \,d\mu(x)\big) \overset \prod_^ \exp\big(a_k \mu(A_k)\big),$

since $a_k$ is the value of $f$ at any point of $A_k$. In the special case where $X = \mathbb$, $\mu$ is Lebesgue measure, and all of the measurable sets $A_k$ are Interval (mathematics), intervals, one can verify that this is equal to the definition given above for that special case. Analogous to Lebesgue integration, the theory of Lebesgue (classical) integrals, the #Type I: Volterra integral, Volterra product integral of any product-integrable function $f$ can be written as the limit of an increasing sequence of Volterra product integrals of product-integrable simple functions.

Taking logarithms of both sides of the above definition, one gets that for any product-integrable simple function $f$:

: $\ln \left(\prod_X \big(1 + f(x) \,d\mu(x)\big) \right) = \ln \left( \prod_^ \exp\big(a_k \mu(A_k)\big) \right) = \sum_^ a_k \mu(A_k) = \int_X f(x) \,d\mu(x) \iff$
: $\prod_X \big(1 + f(x) \,d\mu(x)\big) = \exp \left( \int_X f(x) \,d\mu(x) \right),$

where we used Simple function#Integration of simple functions, the definition of integral for simple functions. Moreover, because continuous functions like $\exp$ Continuous function#Definition in terms of limits of sequences, can be interchanged with limits, and the product integral of any product-integrable function $f$ is equal to the limit of product integrals of simple functions, it follows that the relationship

: $\prod_X \big(1 + f(x) \,d\mu(x)\big) = \exp \left( \int_X f(x) \,d\mu(x) \right)$

holds generally for ''any'' product-integrable $f$. This clearly generalizes the property #Type I: Volterra integral, mentioned above.

The #Type I: Volterra integral, Volterra product integral is Multiplicative function, multiplicative as a set function,Gill, Richard D., Soren Johansen
"A Survey of Product Integration with a View Toward Application in Survival Analysis"
The Annals of Statistics 18, no. 4 (December 1990): 1501—555, p. 1503. which can be shown using the above property. More specifically, given a product-integrable function $f$ one can define a set function $_f$ by defining, for every measurable set $B \subseteq X$,

: $_f(B) \overset \prod_B \big(1 + f(x) \,d\mu(x)\big) \overset \prod_X \big(1 + (f \cdot I_B)(x) \,d\mu(x)\big),$

where $I_B(x)$ denotes the indicator function of $B$. Then for any two Disjoint sets, ''disjoint'' measurable sets $B_1, B_2$ one has

: $\begin _f(B_1 \sqcup B_2) &= \prod_ \big(1 + f(x) \,d\mu(x)\big) \\ &= \exp\left( \int_ f(x) \,d\mu(x) \right) \\ &= \exp\left( \int_ f(x) \,d\mu(x) + \int_ f(x) \,d\mu(x) \right) \\ &= \exp\left( \int_ f(x) \,d\mu(x) \right) \exp\left( \int_ f(x) \,d\mu(x) \right) \\ &= \prod_ (1 + f(x)d \mu(x)) \prod_ (1 + f(x) \,d\mu(x)) \\ &= _f(B_1 ) _f(B_2). \end$

This property can be contrasted with Measure (mathematics), measures, which are Sigma additivity, ''additive'' set functions.

However the #Type I: Volterra integral, Volterra product integral is ''not'' Multiplicative function, multiplicative as a Functional (mathematics), functional. Given two product-integrable functions $f , g$, and a measurable set $A$, it is generally the case that

: $\prod_A \big(1 + (fg)(x) \,d\mu(x)\big) \neq \prod_A \big(1 + f(x) \,d\mu(x)\big) \prod_A \big(1 + g(x) \,d\mu(x)\big).$

Type II: geometric integral

If $X$ is a measure space with Measure (mathematics), measure $\mu$, then for any product-integrable simple function $f(x) = \sum_^n a_k I_(x)$ (i.e. a conical combination of the indicator functions for some Disjoint sets, disjoint Measure (mathematics)#Definition, measurable sets $A_0, A_1, \dots, A_ \subseteq X$), its #Type II: geometric integral, type II product integral is defined to be

: $\prod_X f(x)^ \overset \prod_^ a_k^.$

This can be seen to generalize the definition given above.

Taking logarithms of both sides, we see that for any product-integrable simple function $f$:

: $\ln \left( \prod_X f(x)^ \right) = \sum_^ \ln(a_k) \mu(A_k) = \int_X \ln f(x) \,d\mu (x) \iff \prod_X f(x)^ = \exp\left( \int_X \ln f(x) \,d\mu (x) \right),$

where we have used the Simple function#Integration of simple functions, definition of the Lebesgue integral for simple functions. This observation, analogous to the one already made #Type II: geometric integral, above, allows one to entirely reduce the "Lebesgue integration#Measure theory, Lebesgue theory of #Type II: geometric integral, geometric integrals" to the Lebesgue integration, Lebesgue theory of (classical) integrals. In other words, because continuous functions like $\exp$ and $\ln$ Continuous function#Definition in terms of limits of sequences, can be interchanged with limits, and the product integral of any product-integrable function $f$ is equal to the Limit (mathematics), limit of some increasing sequence of product integrals of simple functions, it follows that the relationship

: $\prod_X f(x)^ = \exp\left( \int_X \ln f(x) \,d\mu(x) \right)$

holds generally for ''any'' product-integrable $f$. This generalizes the property of #Type II: geometric integral, geometric integrals mentioned above.

See also

*List of derivatives and integrals in alternative calculi
*Indefinite product
*Logarithmic derivative
*Ordered exponential
*Fractal derivative

References

External links

Non-Newtonian calculus website
* Richard Gill
''Product Integration''
* Richard Gill
''Product Integral Symbol''
* David Manura

* Tyler Neylon
''Easy bounds for n!''

An Introduction to Multigral (Product) and Dx-less Calculus


* Antonín Slavík
''An introduction to product integration''
* Antonín Slavík
''Henstock–Kurzweil and McShane product integration''

{{DEFAULTSORT:Product Integral
Integrals
Multiplication
Non-Newtonian calculus