<abstract xmlns="http://www.w3.org/1999/xhtml">

Egoroff’s classical theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor’s theorem shows the possibility of controlling the convergence of the sequences of functions on the set of the full measure. Namely, for every sequence of real-valued measurable factions |<italic>fn</italic>}<italic>n</italic>∈ℕ pointwise converging to a function <italic>f</italic> on a measurable set <italic>E</italic>, there exist a decreasing sequence |<italic>δn</italic>}<italic>n</italic>∈ℕ of positive reals converging to 0 and a set <italic>A</italic> ⊆ <italic>E</italic> such that <italic>E</italic> \ <italic>A</italic> is a nullset and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0009_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mi>∞</mi></mrow></msub><mfrac><mrow><mo>|</mo><msub><mi>f</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>|</mo></mrow><mrow><msub><mi>δ</mi><mi>n</mi></msub></mrow></mfrac><mo>=</mo><mn>0</mn><mtext> </mtext><mtext>for</mtext><mtext> </mtext><mtext>all</mtext><mtext> </mtext><mi>x</mi><mo>∈</mo><mi>A</mi><mo>.</mo><mtext> </mtext><mtext>Let</mtext><mtext> </mtext><mi>J</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mtext> </mtext><mo>{</mo><msub><mi>f</mi><mi>n</mi></msub><mo>}</mo><mo stretchy="false">)</mo></mrow></math><tex-math>
{\lim _{n \to + \infty }}\frac{{|{f_n}(x) - f(x)|}}{{{\delta _n}}} = 0\,{\rm{for}}\,{\rm{all}}\,x \in A.\,{\rm{Let}}\,J(A,\,\{ {f_n}\} )</tex-math></alternatives></inline-formula> denote the set of all such sequences |<italic>δn</italic>}<italic>n</italic>∈ℕ. The main results of the paper concern basic properties of sets of all such sequences for a given set <italic>A</italic> and a given sequence of functions. A relationship between pointwise convergence, uniform convergence and the Taylor’s type of convergence is considered.
</abstract>

Egoroff’s classical theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor’s theorem shows the possibility of controlling the convergence of the sequences of functions on the set of the full measure. Namely, for every sequence of real-valued measurable factions |fn}n∈ℕ pointwise converging to a function f on a measurable set E, there exist a decreasing sequence |δn}n∈ℕ of positive reals converging to 0 and a set A ⊆ E such that E \ A is a nullset and limn→+∞|fn(x)−f(x)|δn=0 for all x∈A. Let J(A, {fn})
{\lim _{n \to  + \infty }}\frac{{|{f_n}(x) - f(x)|}}{{{\delta _n}}} = 0\,{\rm{for}}\,{\rm{all}}\,x \in A.\,{\rm{Let}}\,J(A,\,\{ {f_n}\} ) denote the set of all such sequences |δn}n∈ℕ. The main results of the paper concern basic properties of sets of all such sequences for a given set A and a given sequence of functions. A relationship between pointwise convergence, uniform convergence and the Taylor’s type of convergence is considered.

Around Taylor’s Theorem on the Convergence of Sequences of Functions

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Around Taylor\u2019s Theorem on the Convergence of Sequences of Functions"}

Egoroff’s classical theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor’s...

Around Taylor’s Theorem on the Convergence of Sequences of Functions

Published Online: Jan 01, 2022

Page range: 129 - 138

Received: Jun 19, 2021

DOI: https://doi.org/10.2478/tmmp-2021-0009

Keywords
sequences of real-valued functions, convergence

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Around Taylor’s Theorem on the Convergence of Sequences of Functions

Published Online: Jan 01, 2022

Page range: 129 - 138

Received: Jun 19, 2021

DOI: https://doi.org/10.2478/tmmp-2021-0009

Keywordssequences of real-valued functions, convergence

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
sequences of real-valued functions, convergence