Statement
Suppose
is a locally finite Borel measure and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{f_n\}}
is a sequence of measurable functions defined on a measurable set
with
and
a.e. on E.
Then:
Given
we may find a closed subset
such that
and
uniformly on
[1]
Proof
WLOG assume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n \rightarrow f }
for all
since the set of points at which
is a null set. Fix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon>0 }
and for
we define
. Since
are measurable so is their difference. Then since the absolute value of a measurable function is measurable each
is measurable.
Now for fixed
we have that
and
. Therefore using continuity from below we may find a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_n }
such that
.
Now choose
so that
and define
. By countable subadditivity we have that
.
Fix any
. We choose
such that
. Since
if
then
. And by definition if
then
whenever
. Hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n \rightarrow f}
uniformly on
.
Finally, since
is measurable, using HW5 problem 6 there exists a closed set
such that
. Therefore
and
on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_\epsilon }
Corollary
Bounded Convergence Theorem : Let
be a seqeunce of measurable functions bounded by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M }
, supported on a set
and
a.e. Then
[2]
Proof
By assumptions on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n}
,
is measurable, bounded, supported on
for a.e. x. Fix
, then by Egerov we may find a measurable subset
of
such that
and
uniformly on
. Therefore, for sufficiently large
we have that
for all
. Putting this together yields
Since
was arbitrary and
is finite by assumption we are done.
References
- ↑ Stein & Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Chapter 1 §4.3
- ↑ Stein & Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Chapter 2 § 1