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[[Category:Classical measure theory]] ...eveloped in topological spaces, e.g. in Euclidean spaces, the concept of a Baire set coincides with that of a [[Borel set|Borel set]].

...lity of a set. A subset $A$ of a topological space $X$ is said to have the Baire property if there is an open set $U$ such that the symmetric difference $(U A set $A$ has the Baire property if and only if there is a closed set $C$ such that $(C\setminus A)

...te metric space|complete metric space]] is a Baire space. Another class of Baire spaces are [[Locally compact space|locally compact]] [[Hausdorff space|Haus ...y [[Polish space]] $\mathcal{M}$ there is a continuous surjection from the Baire space onto $\mathcal{M}$ (see Theorem 1A.1 of {{Cite|Mo}}).

====Baire category theorem==== Stated by R. Baire {{Cite|Ba1}}.

...self. This result generalizes to any complete metric space, it is called [[Baire category theorem]] ...in $\mathbb R$ a set of the first category can be a set of full (Lebesgue) measure, while there are (Lebesgue) null sets which are residual ({{Cite|vR}}, Th.

The Baire classes are families of real functions on a topological space $X$, indexed * The zero-th Baire class $\mathcal{H}_0$ is the class of continuous functions;

of measure zero on the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $, ...o infinity and zero along all radii that end at points of some set of full measure $ 2 \pi $

ring on which the measure $ \mu $ are the exterior and interior measures, respectively (see [[Measure|Measure]]).

...s related to the [[Baire theorem|Baire theorem]]. Cf. also [[Baire classes|Baire classes]] and {{Cite|Ch}}. |valign="top"|{{Ref|Ox}}|| J.C. Oxtoby, "Measure and category" , Springer (1971).

The [[Baire theorem|Baire Category theorem]] asserts that if $X$ is a complete metric space or a loca |valign="top"|{{Ref|Ox}}|| J.C. Oxtoby, "Measure and category" , Springer (1971) {{MR|0393403}} {{ZBL|0217.09201}}

[[Category:Classical measure theory]] ...$ which coincides with $f$ almost everywhere (with respect to the Lebesgue measure).

...has infinite angular boundary values on a set $E\subset\Gamma$ of positive measure. ...igma$ means that every [[Portion|portion]] of $E$ on $\sigma$ has positive measure. This implies that if the radial boundary values of $f(z)$ on a set $E$ of

The terminology ''Borel measure'' is used by different authors with different meanings: ...5(b) of {{Cite|Ma}} or with Section 1.1 of {{Cite|EG}}) use it for [[Outer measure|outer measures]] $\mu$ on a topological space $X$ for which the Borel sets

...first Baire category (cf. [[Baire classes|Baire classes]]) and of Lebesgue measure zero in $ \mathbf R ^ {n} $. of measure zero that are not $ \sigma $-

===Baire category=== ...n dense subsets in $X$. The terminology is in general used when $X$ is a [[Baire space]]: in such spaces generic sets are dense. When some property $P$ whic

...to be a [[discontinuous function]]. However, according to [[Baire classes|Baire's classification]] it is always a function of the first class and has the [ ...see [[Gradient]]), and of a derivative of a set function with respect to a measure (in particular, with respect to area, volume, etc.). The concept of a deriv

...ces|[a3]]]: Given a set $E \subset ( 0,1 )$ of [[Lebesgue measure|Lebesgue measure]] zero, there is an approximately continuous function $f$ such that $\under

is a [[Baire space|Baire space]], i.e., a space in which open, non-empty subsets are of the second c with the density topology is a Baire space which is not Blumberg. W.A.R. Weiss (see the references of [[#Referen

with measure $ \mu $ ...ifferentiation may be generalized to the case of abstract spaces without a measure [[#References|[3]]].

...dy even of smooth functions. Among such problems one must put those of the measure of a set, the length of curves and the area of surfaces, the primitive and ...are particularly close, their foundations having been laid by E. Borel, R. Baire, H. Lebesgue, and others.

...roved by R. Baire in {{Cite|Ba}} for functions of one real variable (cf. [[Baire theorem]]). ==Relation to Baire classes==

...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;

...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

''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

of positive Lebesgue measure, then $ f ( z) \equiv 0 $. of measure zero on the unit circle $ \Gamma $.

[[Category:Classical measure theory]] ...$\R^n$ with the [[Borel set|Borel σ-algebra]]; $\R^n$ with the [[Lebesgue measure|Lebesgue σ-algebra]].

...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><

...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

...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

is a [[Baire space|Baire space]] in the induced topology (that is, the intersection of any sequence there is a probability measure $ \mu $

[[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}}):

...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.

[[Invariant measure|invariant measure]] and related problems. [[Borel measure|Borel measure]];

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

has linear Hausdorff measure zero, $ \mu ( E) = \mu _ {1} ( E) = 0 $, hence its plane measure $ \mu _ {2} ( CR) = 0 $).

...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 $-

...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 $,

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

probability measure that can also be viewed as the conditional distribution definition of "almost all" is a set of Baire category II. While "almost

...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

...es related to it; the fixed-point theorem for a contraction mapping; the [[Baire theorem]] on the non-emptiness of the intersection of a countable family of ...10]]], Chapt. 22. A considerably wider segment of functional analysis than measure theory depends on topological methods as much as on analytic ones (cf. [[#R