Under Construction
- katherineokikiolu
- 4 days ago
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The Borel Functions
Lecture Notes for Functional Analysis
November 1, 2025
Kate Okikiolu
Computer Support by Hans Lindblad
Computer Apps: WinEdt, Adobe Acrobat
Acknowledgements: Cambridge University 1983 { 1987.
We are interested in the Borel functions on the real line in
order to prove the spectral theorem for self-adjoint operators
on a Hilbert space. However, what we want to discuss here is
the nature of those functions which are the pointwise limits
of the continuous functions. These functions are all Borel
functions. Moreover the Borel functions are closed under
taking pointwise limits.
De nition 1.. If X is a set then a set S of subsets of X is
called a - eld on X if
1. The empty set and the set X are in S.
2. S is closed under taking di erences, meaning that if
A;B are in S then so is A n B.
3. S is closed under countable unions and countable intersections,
meaning that if B1; B2; : : : is a countable sequence
in S, then the union [jBj and the intersection \jBj are in
S.
Lemma 1. S is a - eld on X if it satis es conditions 1
and if in addition
2'. S is closed under taking complements, that is if B is
in S, then so is X n B.
1
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