...roved by R. Baire in {{Cite|Ba}} for functions of one real variable (cf. [[Baire theorem]]).
==Relation to Baire classes==
8 KB (1,260 words) - 15:29, 5 January 2017
...sequence $\{ \mu _ { n } \}$ of countably additive measures (cf. [[Measure|Measure]]) defined on a $\sigma$-algebra $\Sigma$, i.e., $\operatorname { lim } _ {
i) the limit $m$ is a countably additive measure;
5 KB (677 words) - 07:42, 24 November 2023
...ntal theorems of [[Functional analysis|functional analysis]] and [[Measure|measure]] theory.
...n joint continuity; the Orlicz–Pettis theorem (cf. [[Vector measure|Vector measure]]); the kernel theorem for sequence spaces; the Bessaga–Pelczynski theore
6 KB (845 words) - 16:55, 1 July 2020
''measure of a set''
...f a set for some mass distribution throughout the space. The notion of the measure of a set arose in the theory of functions of a real variable in connection
46 KB (7,065 words) - 19:30, 1 January 2021
of positive Lebesgue measure, then $ f ( z) \equiv 0 $.
of measure zero on the unit circle $ \Gamma $.
10 KB (1,496 words) - 08:27, 6 June 2020
[[Category:Classical measure theory]]
...$\R^n$ with the [[Borel set|Borel σ-algebra]]; $\R^n$ with the [[Lebesgue measure|Lebesgue σ-algebra]].
15 KB (2,605 words) - 07:10, 23 September 2012
...mappings. Any continuous image of a Luzin space lying in $Y$ has Lebesgue measure zero and dimension zero. Moreover, it is totally imperfect, that is, it doe
...</TD> <TD valign="top"> N.N. [N.N. Luzin] Lusin, "Sur un problème de M. Baire" ''C.R. Acad. Sci. Paris'' , '''158''' (1914) pp. 1258–1261</TD></TR><
3 KB (449 words) - 09:03, 2 January 2021
...pha$), then $f$ can be chosen to be measurable (respectively, to belong to Baire class $\alpha$).
...align="top">[a26]</td> <td valign="top"> Z. Zahorski, "Sur la classe de Baire des dérivées approximatives d'une fonction quelconque" ''Ann. Soc. Polon
13 KB (1,903 words) - 16:38, 19 March 2023
...unions of them, nowhere-dense sets or sets of the first category, sets of measure zero. A set $\mathfrak{A} \subset \mathfrak{O}$ is regarded as "large" if i
...non-empty open subset of the space $\mathfrak{O}$ or a subset of positive measure. Then one says that this set of objects "cannot be neglected" (but one no l
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is a [[Baire space|Baire space]] in the induced topology (that is, the intersection of any sequence
there is a probability measure $ \mu $
18 KB (2,674 words) - 22:17, 5 June 2020
[[Category:Classical measure theory]]
...le subset of the euclidean space coincides with a Borel set up to a set of measure zero. More precisely (cp. with Proposition 15 of Chapter 3 in {{Cite|Ro}}):
5 KB (796 words) - 19:29, 23 May 2024
...perties of functions are studied on the basis of the idea of the [[Measure|measure]] of a set.
...ury. The foundations of this theory of functions were laid by E. Borel, R. Baire, H. Lebesgue, and others.
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[[Invariant measure|invariant measure]] and related problems.
[[Borel measure|Borel measure]];
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in the sense of every normalized invariant measure of shift dynamical systems (cf. [[Shift dynamical system|Shift dynamical sy
in the sense of every normalized invariant measure) the upper (lower) central exponent of the system of equations in variation
8 KB (1,097 words) - 16:43, 4 June 2020
has linear Hausdorff measure zero, $ \mu ( E) = \mu _ {1} ( E) = 0 $,
hence its plane measure $ \mu _ {2} ( CR) = 0 $).
25 KB (3,728 words) - 09:43, 26 March 2023
...heory was created in the early 20th century by the studies of E. Borel, R. Baire and H. Lebesgue in connection with the measurability of sets. Borel-measura
...ning these functions (cf. [[Baire classes|Baire classes]]; [[Baire theorem|Baire theorem]]). Lebesgue showed that $ B $-
44 KB (6,667 words) - 11:40, 26 March 2023
...ions that studies properties of functions associated with the concept of a measure is usually called the [[Metric theory of functions|metric theory of functio
is a measure defined on the sets $ A \in \mathfrak S $,
34 KB (5,509 words) - 22:06, 28 January 2020
potential richness of the concept of a set of points of measure zero
the young `Normaliens' at the end of the 19th century, including Baire
14 KB (2,139 words) - 18:37, 8 March 2024
probability measure that can also be viewed as the conditional distribution
definition of "almost all" is a set of Baire category II. While "almost
14 KB (2,104 words) - 19:26, 4 March 2024
...morphism]]). Among these are compactness, separability, connectedness, the Baire property, and zero dimensionality. Properties of this type are called topol
...ete metric spaces, preserved under homeomorphisms, is the [[Baire property|Baire property]], on the strength of which each complete metric space without iso
33 KB (5,289 words) - 19:37, 25 March 2023