<s>
In	O
mathematics	O
,	O
more	O
specifically	O
measure	O
theory	O
,	O
there	O
are	O
various	O
notions	O
of	O
the	O
convergence	B-Algorithm
of	I-Algorithm
measures	I-Algorithm
.	O
</s>
<s>
For	O
an	O
intuitive	O
general	O
sense	O
of	O
what	O
is	O
meant	O
by	O
convergence	B-Algorithm
of	I-Algorithm
measures	I-Algorithm
,	O
consider	O
a	O
sequence	O
of	O
measures	O
μn	O
on	O
a	O
space	O
,	O
sharing	O
a	O
common	O
collection	O
of	O
measurable	O
sets	O
.	O
</s>
<s>
In	O
mathematics	O
and	O
statistics	O
,	O
weak	O
convergence	O
is	O
one	O
of	O
many	O
types	O
of	O
convergence	O
relating	O
to	O
the	O
convergence	B-Algorithm
of	I-Algorithm
measures	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
sequence	O
of	O
random	O
variables	O
then	O
Xn	O
is	O
said	O
to	O
converge	O
weakly	O
(	O
or	O
in	O
distribution	O
or	O
in	O
law	O
)	O
to	O
the	O
random	O
variable	O
X:Ω	O
→	O
X	O
as	O
if	O
the	O
sequence	O
of	O
pushforward	O
measures	O
(	O
Xn	O
)	O
∗( P	O
)	O
converges	O
weakly	O
to	O
X∗( P	O
)	O
in	O
the	O
sense	O
of	O
weak	O
convergence	B-Algorithm
of	I-Algorithm
measures	I-Algorithm
on	O
X	O
,	O
as	O
defined	O
above	O
.	O
</s>
<s>
Despite	O
having	O
the	O
same	O
name	O
as	O
weak	O
convergence	O
in	O
the	O
context	O
of	O
functional	O
analysis	O
,	O
weak	O
convergence	B-Algorithm
of	I-Algorithm
measures	I-Algorithm
is	O
actually	O
an	O
example	O
of	O
weak-*	O
convergence	O
.	O
</s>
<s>
Weak	O
convergence	B-Algorithm
of	I-Algorithm
measures	I-Algorithm
is	O
actaully	O
an	O
example	O
of	O
weak-*	O
convergence	O
:	O
</s>
<s>
To	O
illustrate	O
how	O
weak	O
convergence	B-Algorithm
of	I-Algorithm
measures	I-Algorithm
is	O
an	O
example	O
of	O
weak-*	O
topology	O
,	O
we	O
given	O
an	O
example	O
in	O
terms	O
of	O
the	O
closely	O
related	O
notion	O
of	O
vague	O
convergence	O
.	O
</s>
<s>
Applying	O
the	O
definition	O
of	O
weak-*	O
convergence	O
in	O
terms	O
of	O
linear	O
functionals	O
,	O
the	O
characterization	O
of	O
vague	O
convergence	B-Algorithm
of	I-Algorithm
measures	I-Algorithm
is	O
obtained	O
.	O
</s>
<s>
In	O
a	O
probability	O
setting	O
,	O
vague	O
convergence	O
and	O
weak	B-Algorithm
convergence	I-Algorithm
of	I-Algorithm
probability	I-Algorithm
measures	I-Algorithm
are	O
equivalent	O
assuming	O
tightness	O
.	O
</s>
<s>
The	O
weak	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
of	O
probability	O
measures	O
,	O
provided	O
it	O
exists	O
,	O
is	O
a	O
probability	O
measure	O
.	O
</s>
