Selection set from Helly

The selection of Helly's theorem is a mathematical theorem of measure theory and the theory of probability , which provides information about when a series of dimensions or distribution functions a vaguely convergent subsequence has. The connection between the convergence of the measures and the distribution functions is struck by the Helly-Bray theorem . Thus the set provides criteria for the vaguely relative compactness of sequences of sets of measures. The theorem was proven in 1912 by Eduard Helly as an aid in his work on linear functional operators.

statement

A sequence of distribution functions and a sequence of measures are given . ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

Then:

1. If the sequence of distribution functions is uniformly bounded , it has a vaguely convergent subsequence .
2. If the sequence of measures is limited, i.e. if the sequence of the norms of total variation is a limited sequence , then it has a vaguely convergent subsequence . ${\displaystyle (\|\mu _{n}\|)_{n\in \mathbb {N} }}$

Evidence sketch

The naming as a “selection sentence” is derived from the proof technique. The proof uses a combination of Bolzano-Weierstrass theorem and a diagonal argument . To do this, consider a count of . Since the sequence of the distribution functions is uniformly bounded, the real sequence is also bounded and contains a convergent subsequence that is created by a suitable selection of distribution functions. The selection, evaluated in , is again restricted and thus contains a convergent partial sequence that was again created by selection. ${\displaystyle (q_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle (F_{n}(q_{1}))_{n\in \mathbb {N} }}$ ${\displaystyle (F_{1n}(q_{1}))_{n\in \mathbb {N} }}$ ${\displaystyle q_{2}}$ ${\displaystyle (F_{1n}(q_{2}))_{n\in \mathbb {N} }}$ ${\displaystyle (F_{2n}(q_{2}))_{n\in \mathbb {N} }}$

If this procedure is continued, it can be shown that the diagonal sequence of the distribution functions converges at every point and that a distribution function can be constructed from these point-by-point limit values . So the method delivers a partial sequence of the distribution function sequence. The vague convergence of the distribution functions is deduced from direct recalculation at the points of continuity of the constructed function. ${\displaystyle (F_{nn})_{n\in \mathbb {N} }}$ ${\displaystyle (q_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {R} }$

The proof of the second statement follows directly from Helly-Bray's theorem .

Individual evidence

1. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 392.

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 380-392 , doi : 10.1007 / 978-3-540-89728-6 .
• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 256-265 , doi : 10.1007 / 978-3-642-36018-3 .