<s>
Otherwise	O
,	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
can	O
refer	O
to	O
either	O
global	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
or	O
local	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
,	O
depending	O
on	O
the	O
author	O
.	O
</s>
<s>
Global	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
implies	O
local	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
.	O
</s>
<s>
The	O
converse	O
,	O
however	O
,	O
is	O
false	O
;	O
i.e.	O
,	O
local	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
is	O
strictly	O
weaker	O
than	O
global	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
,	O
in	O
general	O
.	O
</s>
<s>
If	O
,	O
however	O
,	O
or	O
,	O
more	O
generally	O
,	O
if	O
f	O
and	O
all	O
the	O
fn	O
vanish	O
outside	O
some	O
set	O
of	O
finite	O
measure	O
,	O
then	O
the	O
distinction	O
between	O
local	O
and	O
global	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
disappears	O
.	O
</s>
<s>
The	O
assumption	O
of	O
σ-finiteness	O
is	O
not	O
necessary	O
in	O
the	O
case	O
of	O
global	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
.	O
</s>
<s>
Fatou	O
's	O
lemma	O
and	O
the	O
monotone	O
convergence	O
theorem	O
hold	O
if	O
almost	O
everywhere	O
convergence	O
is	O
replaced	O
by	O
(	O
local	O
or	O
global	O
)	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
.	O
</s>
<s>
If	O
μ	O
is	O
σ-finite	O
,	O
Lebesgue	O
's	O
dominated	O
convergence	O
theorem	O
also	O
holds	O
if	O
almost	O
everywhere	O
convergence	O
is	O
replaced	O
by	O
(	O
local	O
or	O
global	O
)	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
.	O
</s>
<s>
There	O
is	O
a	O
topology	O
,	O
called	O
the	O
topology	O
of	O
(	O
local	O
)	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
,	O
on	O
the	O
collection	O
of	O
measurable	O
functions	O
from	O
X	O
such	O
that	O
local	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
corresponds	O
to	O
convergence	O
on	O
that	O
topology	O
.	O
</s>
<s>
still	O
defines	O
a	O
metric	O
that	O
generates	O
the	O
global	O
convergence	B-Algorithm
in	I-Algorithm
measure	I-Algorithm
.	O
</s>
