In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
Fubini's theorem is a result that gives conditions under which it is possible to compute a
double integral
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
by using an
iterated integral
In multivariable calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example f(x,y) or f(x,y,z)) in a way that each of the integrals considers some of the variables as given constants. ...
, introduced by
Guido Fubini
Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric.
Life
Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, wh ...
in 1907. One may switch the
order of integration
In statistics, the order of integration, denoted ''I''(''d''), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series.
Integration of order ''d''
A time seri ...
if the double integral yields a finite answer when the integrand is replaced by its absolute value.
Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands. Tonelli's theorem, introduced by
Leonida Tonelli
Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calc ...
in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains.
A related theorem is often called Fubini's theorem for infinite series, which states that if
is a doubly-indexed sequence of real numbers, and if
is absolutely convergent, then
Although Fubini's theorem for infinite series is a special case of the more general Fubini's theorem, it is not appropriate to characterize it as a logical consequence of Fubini's theorem. This is because some properties of measures, in particular sub-additivity, are often proved using Fubini's theorem for infinite series. In this case, Fubini's general theorem is a logical consequence of Fubini's theorem for infinite series.
History
The special case of Fubini's theorem for continuous functions on a product of closed bounded subsets of real vector spaces was known to
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in the 18th century. extended this to bounded measurable functions on a product of intervals. conjectured that the theorem could be extended to functions that were integrable rather than bounded, and this was proved by . gave a variation of Fubini's theorem that applies to non-negative functions rather than integrable functions.
Product measures
If ''X'' and ''Y'' are
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
s with measures, there are several natural ways to define a
product measure In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two ...
on their product.
The product ''X'' × ''Y'' of measure spaces (in
the sense of category theory) has as its measurable sets the
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
generated by the products ''A'' × ''B'' of measurable subsets of ''X'' and ''Y''.
A measure ''μ'' on ''X'' × ''Y'' is called a product measure if ''μ''(''A'' × ''B'') = ''μ''
1(''A'')''μ''
2(''B'') for measurable subsets ''A'' ⊂ ''X'' and ''B'' ⊂ ''Y'' and measures ''µ''
1 on ''X'' and ''µ''
2 on ''Y''. In general there may be many different product measures on ''X'' × ''Y''. Fubini's theorem and Tonelli's theorem both need technical conditions to avoid this complication; the most common way is to assume all measure spaces are
σ-finite, in which case there is a unique product measure on ''X''×''Y''. There is always a unique maximal product measure on ''X'' × ''Y'', where the measure of a measurable set is the inf of the measures of sets containing it that are countable unions of products of measurable sets. The maximal product measure can be constructed by applying
Carathéodory's extension theorem
In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets ''R'' of a given set ''Ω'' can be extended to a measure on the σ- ...
to the additive function μ such that ''μ''(''A'' × ''B'') = ''μ''
1(''A'')''μ''
2(''B'') on the ring of sets generated by products of measurable sets. (Carathéodory's extension theorem gives a measure on a measure space that in general contains more measurable sets than the measure space ''X'' × ''Y'', so strictly speaking the measure should be restricted to the
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
generated by the products ''A'' × ''B'' of measurable subsets of ''X'' and ''Y''.)
The product of two
complete measure spaces is not usually complete. For example, the product of the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
on the unit interval ''I'' with itself is not the Lebesgue measure on the square ''I'' × ''I''. There is a variation of Fubini's theorem for complete measures, which uses the completion of the product of measures rather than the uncompleted product.
For integrable functions
Suppose ''X'' and ''Y'' are
σ-finite measure spaces, and suppose that ''X'' × ''Y'' is given the product measure (which is unique as ''X'' and ''Y'' are σ-finite). Fubini's theorem states that if ''f'' is ''X'' × ''Y'' integrable, meaning that ''f'' is a
measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
and
then
The first two integrals are iterated integrals with respect to two measures, respectively, and the third is an integral with respect to the product measure. The partial integrals
and
need not be defined everywhere, but this does not matter as the points where they are not defined form a set of measure 0.
If the above integral of the absolute value is not finite, then the two iterated integrals may have different values. See
below for an illustration of this possibility.
The condition that ''X'' and ''Y'' are σ-finite is usually harmless because in practice almost all measure spaces one wishes to use Fubini's theorem for are σ-finite.
Fubini's theorem has some rather technical extensions to the case when ''X'' and ''Y'' are not assumed to be σ-finite . The main extra complication in this case is that there may be more than one product measure on ''X''×''Y''. Fubini's theorem continues to hold for the maximal product measure, but can fail for other product measures. For example, there is a product measure and a non-negative measurable function ''f'' for which the double integral of , ''f'', is zero but the two iterated integrals have different values; see the section on counterexamples below for an example of this. Tonelli's theorem and the Fubini–Tonelli theorem (stated below) can fail on non σ-finite spaces even for the maximal product measure.
Tonelli's theorem for non-negative measurable functions
Tonelli's theorem (named after
Leonida Tonelli
Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calc ...
) is a successor of Fubini's theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumption that
has a finite integral is replaced by the assumption that
is a non-negative measurable function.
Tonelli's theorem states that if (''X'', ''A'', μ) and (''Y'', ''B'', ν) are
σ-finite measure spaces, while ''f'' from ''X''×''Y'' to
,∞is non-negative measurable function, then
A special case of Tonelli's theorem is in the interchange of the summations, as in
, where
are non-negative for all ''x'' and ''y''. The crux of the theorem is that the interchange of order of summation holds even if the series diverges. In effect, the only way a change in order of summation can change the sum is when there exist some subsequences that diverge to
and others diverging to
. With all elements non-negative, this does not happen in the stated example.
Without the condition that the measure spaces are σ-finite it is possible for all three of these integrals to have different values.
Some authors give generalizations of Tonelli's theorem to some measure spaces that are not σ-finite but these generalizations often add conditions that immediately reduce the problem to the σ-finite case. For example, one could take the σ-algebra on ''A''×''B'' to be that generated by the product of subsets of finite measure, rather than that generated by all products of measurable subsets, though this has the undesirable consequence that the projections from the product to its factors ''A'' and ''B'' are not measurable. Another way is to add the condition that the support of ''f'' is contained in a countable union of products of sets of finite measure. gives some rather technical extensions of Tonelli's theorem to some non σ-finite spaces. None of these generalizations have found any significant applications outside abstract measure theory, largely because almost all measure spaces of practical interest are σ-finite.
Fubini–Tonelli theorem
Combining Fubini's theorem with Tonelli's theorem gives
the Fubini–Tonelli theorem (often just called Fubini's theorem), which states that if
and
are
σ-finite measure
In mathematics, a positive (or signed) measure ''μ'' defined on a ''σ''-algebra Σ of subsets of a set ''X'' is called a finite measure if ''μ''(''X'') is a finite real number (rather than ∞), and a set ''A'' in Σ is of finite measur ...
spaces, and if
is a measurable function, then
Besides if any one of these integrals is finite, then
The absolute value of
in the conditions above can be replaced by either the positive or the negative part of
; these forms include Tonelli's theorem as a special case as the negative part of a non-negative function is zero and so has finite integral. Informally all these conditions say that the double integral of
is well defined, though possibly infinite.
The advantage of the Fubini–Tonelli over Fubini's theorem is that the repeated integrals of
may be easier to study than the double integral. As in Fubini's theorem, the single integrals may fail to be defined on a measure 0 set.
For complete measures
The versions of Fubini's and Tonelli's theorems above do not apply to integration on the product of the real line
with itself with Lebesgue measure. The problem is that Lebesgue measure on
is not the product of Lebesgue measure on
with itself, but rather the completion of this: a product of two complete measure spaces
and
is not in general complete. For this reason one sometimes uses versions of Fubini's theorem for complete measures: roughly speaking one just replaces all measures by their completions. The various versions of Fubini's theorem are similar to the versions above, with the following minor differences:
*Instead of taking a product
of two measure spaces, one takes the completion of some product.
*If
is measurable on the completion of
then its restrictions to vertical or horizontal lines may be non-measurable for a measure zero subset of lines, so one has to allow for the possibility that the vertical or horizontal integrals are undefined on a set of measure 0 because they involve integrating non-measurable functions. This makes little difference, because they can already be undefined due to the functions not being integrable.
*One generally also assumes that the measures on
and
are complete, otherwise the two partial integrals along vertical or horizontal lines may be well-defined but not measurable. For example, if
is the characteristic function of a product of a measurable set and a non-measurable set contained in a measure 0 set then its single integral is well defined everywhere but non-measurable.
Proofs
Proofs of the Fubini and Tonelli theorems are necessarily somewhat technical, as they have to use a hypothesis related to σ-finiteness. Most proofs involve building up to the full theorems by proving them for increasingly complicated functions with the steps as follows.
# Use the fact that the measure on the product is a product measure to prove the theorems for the characteristic functions of rectangles.
# Use the condition that the spaces are σ-finite (or some related condition) to prove the theorem for the characteristic functions of measurable sets. This also covers the case of simple measurable functions (measurable functions taking only a finite number of values).
# Use the condition that the functions are measurable to prove the theorems for positive measurable functions by approximating them by simple measurable functions. This proves Tonelli's theorem.
# Use the condition that the functions are integrable to write them as the difference of two positive integrable functions, and apply Tonelli's theorem to each of these. This proves Fubini's theorem.
Riemann integrals
For
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
s, Fubini's theorem is proven by refining the partitions along the x-axis and y-axis as to create a joint partition of the form