Real Analysis Exchange

Oscillation and ω-Primitives

J. Ewert and S. P. Ponomarev

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Abstract

We extend the results of [2], [6] in the case of topological spaces. It is shown that given an upper semicontinuous (USC) function $f:X\to [0,\infty)$ where $X$ is a massive first countable $T_1$-space satisfying some "neighborhood conditions", there exists $F:X\to [0, \infty)$ whose oscillation equals $f$ everywhere on $X$ (Theorem 2.1). The analogous result holds for USC functions $f:X\to [0, \infty]$ if, in addition, $X$ is a normal space (Theorem 2.4). A special metrizability criterion is established (Theorem 1.1). This is to show, by exhibiting corresponding examples, that the neighborhood conditions and massiveness do not imply that $X$ is Baire or metrizable. Among other related topics, sequences of $\omega$-primitives are discussed.

Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 687-702.

Dates
First available in Project Euclid: 27 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1214571360

Mathematical Reviews number (MathSciNet)
MR1844386

Zentralblatt MATH identifier
1025.26002

Subjects
Primary: 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} 54C30: Real-valued functions [See also 26-XX] 54C99: None of the above, but in this section

Keywords
first countable space $\sigma$-discrete set Teichm\"uller-Tukey's lemma upper semicontinuous function massive space oscillation $\omega$-primitive quasi-uniform convergence

Citation

Ewert, J.; Ponomarev, S. P. Oscillation and ω-Primitives. Real Anal. Exchange 26 (2000), no. 2, 687--702. https://projecteuclid.org/euclid.rae/1214571360


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