A NOTE ON GENERALIZED EGOROV’S THEOREM

Tomasz Weiss

Abstract


We prove  that the following generalized version  of Egorov’s  theorem is independent  from the ZFC  axioms  of the set theory.

Let  {fn }nω , fn :  0, 1  → R, be  a sequence  of functions  (not  nec- essarily  measurable) converging   pointwise  to zero  for every  x ∈   0, 1 . Then   for  every  ε  >  0,  there   are  a  set  A  ⊂   0, 1  of Lebesgue  outer measure  m∗  > 1 ε and  a sequence  of integers {nk }kω   with  {fnk }kω converging  uniformly on A.

 


Full Text:

PDF

References


Tomek Bartoszyn´ski and Haim Judah, Set theory: on the structure of the real line, A.K. Peters, Massachussets, 1995.

Lev Bukovsk´y, The Structure od the Real Line, Monografie Matematyczne, Instytut

Matematyczny PAN, Vol. 71, New Series, Birkah¨auser, 2011.

Fausto Di Biase, Aleksander Stokolos, Olof Svenson, and Tomasz Weiss, On the sharp- ness of the Stolz approach, Ann. Acad. Sci. Fenn. Math., 31: 47–59, 2007.

Micha l Korch, Generalized Egorov’s statement for ideals, to be published in Real Anal-

ysis Exchange, 2017.

Roberto Pinciroli, On the independence of a generalized statement of Egoroff ’s theorem from ZFC after T. Weiss, Real Anal. Exchange, 32 (1): 225–232, 2006/2007.

Miroslav Repick´y, On generalized Egoroff ’s theorem, Tatra Mt. Math. Publ., 44: 81–96, 2008.

Tomasz Weiss, A note on generalized Egorov’s theorem, preprint, 2004.


Refbacks

  • There are currently no refbacks.