Real Analysis Exchange

The ℐ-almost constant convergence of sequences of real functions.

Tomasz Natkaniec

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Abstract

Let \(T\) be an infinite set and \(\mathcal{I}\) be a fixed ideal on \(T\). We introduce and study the notion of almost constant convergence of sequences \(\{ f_t\colon t\in T\}\) of real functions with respect to the ideal \(\mathcal{I}\). This notion generalizes discrete convergence, transfinite convergence, and \(\omega_2\)-convergence. In particular, we consider the question when a given family of functions (e.g., continuous, Baire class 1, Borel measurable, Lebesgue measurable, or functions with the Baire property) is closed with respect to this kind of convergence.

Article information

Source
Real Anal. Exchange, Volume 28, Number 2 (2002), 481-492.

Dates
First available in Project Euclid: 20 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.rae/1184963811

Mathematical Reviews number (MathSciNet)
MR2010331

Subjects
Primary: 26A03: Foundations: limits and generalizations, elementary topology of the line
Secondary: 03E35: Consistency and independence results 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]

Keywords
ideal of sets; additivity; covering; \(\I\)-convergence; $\I$-almost constant family; point-$\I$-disjoint family 0-1 set.

Citation

Natkaniec, Tomasz. The ℐ-almost constant convergence of sequences of real functions. Real Anal. Exchange 28 (2002), no. 2, 481--492. https://projecteuclid.org/euclid.rae/1184963811


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