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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v0960001.png" /> be a locally compact Hausdorff space. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v0960002.png" /> is second countable (i.e. there is a countable base). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v0960003.png" /> is a Polish space (there exists a complete separable metrization). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v0960004.png" /> be the Borel field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v0960005.png" /> (cf. [[Borel field of sets|Borel field of sets]]), generated by the (set of open subsets of the) topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v0960006.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v0960007.png" /> be the ring of all relatively compact elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v0960008.png" />, the ring of bounded Borel sets. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v0960009.png" /> be the collection of all Borel measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600010.png" /> (cf. [[Borel measure|Borel measure]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600011.png" /> be the space of real-valued functions of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600012.png" />. A sequence of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600013.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600014.png" /> if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600015.png" />,
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$#C+1 = 38 : ~/encyclopedia/old_files/data/V096/V.0906000 Vague topology
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{{TEX|done}}
  
The topology thus obtained on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600017.png" /> is called the vague topology. If (*) is required to hold for all bounded continuous functions, one obtains the weak topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600018.png" />. Thus, the vague topology is weaker than the weak topology. The difference is illustrated by the observation that a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600019.png" /> is relatively compact in the vague topology if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600020.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600021.png" /> and is relatively compact in the weak topology if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600022.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600024.png" />.
+
Let  $  X $
 +
be a locally compact Hausdorff space. Assume that  $  X $
 +
is second countable (i.e. there is a countable base). Then  $  X $
 +
is a Polish space (there exists a complete separable metrization). Let  $  \mathfrak X $
 +
be the Borel field of  $  X $(
 +
cf. [[Borel field of sets|Borel field of sets]]), generated by the (set of open subsets of the) topology of  $  X $.  
 +
Let  $  \mathfrak B $
 +
be the ring of all relatively compact elements of  $  \mathfrak X $,
 +
the ring of bounded Borel sets. Let  $  \mathfrak M $
 +
be the collection of all Borel measures on  $  X $(
 +
cf. [[Borel measure|Borel measure]]). Let  $  \mathfrak F _ {c} $
 +
be the space of real-valued functions of compact support on  $  X $.  
 +
A sequence of elements  $  \mu _ {n} \in \mathfrak M $
 +
converges to  $  \mu \in \mathfrak M $
 +
if for all $  f \in \mathfrak F _ {c} $,
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600025.png" /> be the set of all integer-valued elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600026.png" />, i.e. those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600027.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600028.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600029.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600030.png" /> is vaguely closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600031.png" />. Both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600033.png" /> are Polish in the vague topology.
+
$$ \tag{* }
 +
\int\limits _ { X } f( x) \mu _ {n} ( dx )  = \
 +
\int\limits _ { X } f( x) \mu ( dx) .
 +
$$
  
If a sequence of real random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600034.png" /> on a probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600035.png" /> converges in probability (cf. [[Convergence in probability|Convergence in probability]]) to a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600036.png" />, then their associated measures converge vaguely. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600037.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096000/v09600038.png" />-almost surely constant, the converse also holds.
+
The topology thus obtained on  $  \mathfrak M $
 +
is called the vague topology. If (*) is required to hold for all bounded continuous functions, one obtains the weak topology on  $  \mathfrak M $.
 +
Thus, the vague topology is weaker than the weak topology. The difference is illustrated by the observation that a subset  $  {\mathcal M} \subset  \mathfrak M $
 +
is relatively compact in the vague topology if and only if  $  \mu ( B) < \infty $
 +
for all  $  \mu \in {\mathcal M} $
 +
and is relatively compact in the weak topology if and only if  $  \mu ( X) < \infty $
 +
for all  $  \mu \in {\mathcal M} $
 +
and  $  \inf _ {B \in \mathfrak B }  \sup _ {\mu \in {\mathcal M} }  \mu ( X \setminus  B) = 0 $.
 +
 
 +
Let  $  \mathfrak N $
 +
be the set of all integer-valued elements of  $  \mathfrak M $,
 +
i.e. those  $  \mu \in \mathfrak M $
 +
for which  $  \mu ( B) \in \{ 0, 1, 2, .  . . \} $
 +
for all  $  B \in \mathfrak B $.  
 +
Then  $  \mathfrak N $
 +
is vaguely closed in  $  \mathfrak M $.
 +
Both  $  \mathfrak N $
 +
and  $  \mathfrak M $
 +
are Polish in the vague topology.
 +
 
 +
If a sequence of real random variables  $  Y _ {n} $
 +
on a probability space $  ( \Omega , {\mathcal A} , {\mathsf P} ) $
 +
converges in probability (cf. [[Convergence in probability|Convergence in probability]]) to a random variable $  Y $,  
 +
then their associated measures converge vaguely. If $  Y $
 +
is $  {\mathsf P} $-
 +
almost surely constant, the converse also holds.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bauer,  "Probability theory and elements of measure theory" , Holt, Rinehart &amp; Winston  (1972)  pp. §7.7  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Kallenberg,  "Random measures" , Akademie Verlag &amp; Acad. Press  (1986)  pp. Chapt. 15</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Grandell,  "Doubly stochastic Poisson processes" , Springer  (1976)  pp. Appendix</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Bourbaki,  "Intégration" , ''Eléments de mathématiques'' , Hermann  (1965)  pp. Chapt. 1–4, §3.9</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bauer,  "Probability theory and elements of measure theory" , Holt, Rinehart &amp; Winston  (1972)  pp. §7.7  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Kallenberg,  "Random measures" , Akademie Verlag &amp; Acad. Press  (1986)  pp. Chapt. 15</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Grandell,  "Doubly stochastic Poisson processes" , Springer  (1976)  pp. Appendix</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Bourbaki,  "Intégration" , ''Eléments de mathématiques'' , Hermann  (1965)  pp. Chapt. 1–4, §3.9</TD></TR></table>

Revision as of 08:27, 6 June 2020


Let $ X $ be a locally compact Hausdorff space. Assume that $ X $ is second countable (i.e. there is a countable base). Then $ X $ is a Polish space (there exists a complete separable metrization). Let $ \mathfrak X $ be the Borel field of $ X $( cf. Borel field of sets), generated by the (set of open subsets of the) topology of $ X $. Let $ \mathfrak B $ be the ring of all relatively compact elements of $ \mathfrak X $, the ring of bounded Borel sets. Let $ \mathfrak M $ be the collection of all Borel measures on $ X $( cf. Borel measure). Let $ \mathfrak F _ {c} $ be the space of real-valued functions of compact support on $ X $. A sequence of elements $ \mu _ {n} \in \mathfrak M $ converges to $ \mu \in \mathfrak M $ if for all $ f \in \mathfrak F _ {c} $,

$$ \tag{* } \int\limits _ { X } f( x) \mu _ {n} ( dx ) = \ \int\limits _ { X } f( x) \mu ( dx) . $$

The topology thus obtained on $ \mathfrak M $ is called the vague topology. If (*) is required to hold for all bounded continuous functions, one obtains the weak topology on $ \mathfrak M $. Thus, the vague topology is weaker than the weak topology. The difference is illustrated by the observation that a subset $ {\mathcal M} \subset \mathfrak M $ is relatively compact in the vague topology if and only if $ \mu ( B) < \infty $ for all $ \mu \in {\mathcal M} $ and is relatively compact in the weak topology if and only if $ \mu ( X) < \infty $ for all $ \mu \in {\mathcal M} $ and $ \inf _ {B \in \mathfrak B } \sup _ {\mu \in {\mathcal M} } \mu ( X \setminus B) = 0 $.

Let $ \mathfrak N $ be the set of all integer-valued elements of $ \mathfrak M $, i.e. those $ \mu \in \mathfrak M $ for which $ \mu ( B) \in \{ 0, 1, 2, . . . \} $ for all $ B \in \mathfrak B $. Then $ \mathfrak N $ is vaguely closed in $ \mathfrak M $. Both $ \mathfrak N $ and $ \mathfrak M $ are Polish in the vague topology.

If a sequence of real random variables $ Y _ {n} $ on a probability space $ ( \Omega , {\mathcal A} , {\mathsf P} ) $ converges in probability (cf. Convergence in probability) to a random variable $ Y $, then their associated measures converge vaguely. If $ Y $ is $ {\mathsf P} $- almost surely constant, the converse also holds.

References

[a1] H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) pp. §7.7 (Translated from German)
[a2] O. Kallenberg, "Random measures" , Akademie Verlag & Acad. Press (1986) pp. Chapt. 15
[a3] J. Grandell, "Doubly stochastic Poisson processes" , Springer (1976) pp. Appendix
[a4] N. Bourbaki, "Intégration" , Eléments de mathématiques , Hermann (1965) pp. Chapt. 1–4, §3.9
How to Cite This Entry:
Vague topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vague_topology&oldid=49102