In mathematical analysis, Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration 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 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.
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 in the 18th century. extended this to bounded measurable functions on a product of intervals. Levi 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.
If X and Y are measure spaces with measures, there are several natural ways to define a product measure on their product.
The product X × Y of measure spaces (in ) has as its measurable sets the σ-algebra generated by the products A × B of measurable subsets of X and Y.
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In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the X-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by We now wish to construct some two-dimensional Lebesgue measure on the plane as a product 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 measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.
The goal of this thesis is to study continuous-domain inverse problems for the reconstruction of sparse signals and to develop efficient algorithms to solve such problems computationally. The task is to recover a signal of interest as a continuous function ...
Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calcul ...
We define p-adic BPS or pBPS invariants for moduli spaces M-beta,M-chi of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field F. Our definition relies on a canonical measure mu can on the F-analyt ...
Dans ce cours on définira et étudiera la notion de mesure et d'intégrale contre une mesure dans un cadre général, généralisant ce qui a été fait en Analyse IV dans le cas réel.
On verra aussi quelques
The course is based on Durrett's text book
Probability: Theory and Examples.
It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.
This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex