Giuseppe Vitali
Encyclopedia
Giuseppe Vitali was an Italian
mathematician
who worked in several branches of mathematical analysis
.
of real number
s, see Vitali set
. His covering theorem is a fundamental result in measure theory. He also proved several theorems concerning convergence of sequences of measurable
and holomorphic
functions. Vitali convergence theorem
generalizes Lebesgue's
dominated convergence theorem
. Another theorem bearing his name gives a sufficient condition for the uniform convergence of a sequence of holomorphic functions on an open domain
⊂ℂ to a holomorphic function on 
This result has been generalized to normal families
of meromorphic functions, holomorphic function
s of several complex variables
, and so on.
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who worked in several branches of mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
.
Mathematical contributions
Vitali was the first to give an example of a non-measurable subsetNon-measurable set
In mathematics, a non-measurable set is a set whose structure is so complicated that it cannot be assigned any meaningful measure. Such sets are constructed to shed light on the notions of length, area and volume in formal set theory....
of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, see Vitali set
Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...
. His covering theorem is a fundamental result in measure theory. He also proved several theorems concerning convergence of sequences of measurable
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
and holomorphic
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
functions. Vitali convergence theorem
Vitali convergence theorem
In mathematics, the Vitali convergence theorem is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is useful when a dominating function cannot be found for the sequence of functions in question; when such a dominating function can be found, Lebesgue's theorem...
generalizes Lebesgue's
Henri Lebesgue
Henri Léon Lebesgue was a French mathematician most famous for his theory of integration, which was a generalization of the seventeenth century concept of integration—summing the area between an axis and the curve of a function defined for that axis...
dominated convergence theorem
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
. Another theorem bearing his name gives a sufficient condition for the uniform convergence of a sequence of holomorphic functions on an open domain
⊂ℂ to a holomorphic function on This result has been generalized to normal familiesNormal family
In mathematics, with special application to complex analysis, a normal family is a pre-compact family of continuous functions. Informally, this means that the functions in the family are not exceedingly numerous or widely spread out; rather, they stick together in a relatively "compact" manner...
of meromorphic functions, holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s of several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
, and so on.
See also
- Vitali convergence theoremVitali convergence theoremIn mathematics, the Vitali convergence theorem is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is useful when a dominating function cannot be found for the sequence of functions in question; when such a dominating function can be found, Lebesgue's theorem...
- Vitali covering theorem
- Vitali-Hahn-Saks theorem
- Vitali setVitali setIn mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...