[[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]].
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...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)
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...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}}).
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====Baire category theorem====
Stated by R. Baire {{Cite|Ba1}}.
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...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.
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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;
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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 $
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ring on which the measure $ \mu $
are the exterior and interior measures, respectively (see [[Measure|Measure]]).
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...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).
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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}}
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[[Category:Classical measure theory]]
...$ which coincides with $f$ almost everywhere (with respect to the Lebesgue measure).
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...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
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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
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...first Baire category (cf. [[Baire classes|Baire classes]]) and of Lebesgue measure zero in $ \mathbf R ^ {n} $.
of measure zero that are not $ \sigma $-
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===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
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...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
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...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
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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
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with measure $ \mu $
...ifferentiation may be generalized to the case of abstract spaces without a measure [[#References|[3]]].
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...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.
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