In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Helly's selection theorem (also called the ''Helly selection principle'') states that a uniformly bounded sequence of monotone real functions admits a
convergent subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
.
In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions.
It is named for the
Austria
Austria, formally the Republic of Austria, is a landlocked country in Central Europe, lying in the Eastern Alps. It is a federation of nine Federal states of Austria, states, of which the capital Vienna is the List of largest cities in Aust ...
n
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Eduard Helly.
A more general version of the theorem asserts compactness of the space BV
loc of functions locally of
bounded total variation that are
uniformly bounded
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family.
...
at a point.
The theorem has applications throughout
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 ( ...
. In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, the result implies compactness of a
tight family of measures.
Statement of the theorem
Let (''f''
''n'')
''n'' ∈ N be a sequence of increasing functions mapping a real interval I into the real line R,
and suppose that it is uniformly bounded: there are ''a,b'' ∈ R such that ''a'' ≤ ''f''
''n'' ≤ ''b'' for every ''n'' ∈ N.
Then the sequence (''f''
''n'')
''n'' ∈ N admits a pointwise convergent subsequence.
Proof
Step 1. An increasing function ''f'' on an interval I has at most countably many points of discontinuity.
Let
, i.e. the set of discontinuities, then since ''f'' is increasing, any ''x'' in A ''satisfies''
, where
,
, hence by discontinuity,