Term by term integration.
$begingroup$
How do I go about proving this statement?
If $$sum_{k=1}^infty f_k(x)$$is a series of nonnegative measurable functions and $$sum_{k=1}^infty left(int_Ef_k(x)dxright)$$ converges, then $$sum_
{k=1}^infty f_k(x)$$ converges almost everywhere and $$int_Eleft(sum_{k=1}^infty f_k(x)right)dx=sum_{k=1}^inftyleft(int_E f_k(x)dxright)$$
measure-theory lebesgue-integral
asked Nov 28 '18 at 0:39
ICanMakeYouHateMEICanMakeYouHateME
154
$endgroup$
add a comment |
$begingroup$
How do I go about proving this statement?
If $$sum_{k=1}^infty f_k(x)$$is a series of nonnegative measurable functions and $$sum_{k=1}^infty left(int_Ef_k(x)dxright)$$ converges, then $$sum_
{k=1}^infty f_k(x)$$ converges almost everywhere and $$int_Eleft(sum_{k=1}^infty f_k(x)right)dx=sum_{k=1}^inftyleft(int_E f_k(x)dxright)$$
measure-theory lebesgue-integral
asked Nov 28 '18 at 0:39
ICanMakeYouHateMEICanMakeYouHateME
154
$endgroup$
$begingroup$
When asking questions here it is important to describe some of your effort in attacking the problem. I'll a partial answer and leave something for you to finish.
$endgroup$
– RRL
Nov 28 '18 at 7:44
add a comment |
$begingroup$
How do I go about proving this statement?
If $$sum_{k=1}^infty f_k(x)$$is a series of nonnegative measurable functions and $$sum_{k=1}^infty left(int_Ef_k(x)dxright)$$ converges, then $$sum_
{k=1}^infty f_k(x)$$ converges almost everywhere and $$int_Eleft(sum_{k=1}^infty f_k(x)right)dx=sum_{k=1}^inftyleft(int_E f_k(x)dxright)$$
measure-theory lebesgue-integral
asked Nov 28 '18 at 0:39
ICanMakeYouHateMEICanMakeYouHateME
154
$endgroup$
How do I go about proving this statement?
If $$sum_{k=1}^infty f_k(x)$$is a series of nonnegative measurable functions and $$sum_{k=1}^infty left(int_Ef_k(x)dxright)$$ converges, then $$sum_
{k=1}^infty f_k(x)$$ converges almost everywhere and $$int_Eleft(sum_{k=1}^infty f_k(x)right)dx=sum_{k=1}^inftyleft(int_E f_k(x)dxright)$$
measure-theory lebesgue-integral
measure-theory lebesgue-integral
asked Nov 28 '18 at 0:39
ICanMakeYouHateMEICanMakeYouHateME
154
asked Nov 28 '18 at 0:39
ICanMakeYouHateMEICanMakeYouHateME
154
asked Nov 28 '18 at 0:39
ICanMakeYouHateMEICanMakeYouHateME
154
asked Nov 28 '18 at 0:39
ICanMakeYouHateMEICanMakeYouHateME
154
asked Nov 28 '18 at 0:39
ICanMakeYouHateMEICanMakeYouHateME
154
154
$begingroup$
When asking questions here it is important to describe some of your effort in attacking the problem. I'll a partial answer and leave something for you to finish.
$endgroup$
– RRL
Nov 28 '18 at 7:44
add a comment |
$begingroup$
When asking questions here it is important to describe some of your effort in attacking the problem. I'll a partial answer and leave something for you to finish.
$endgroup$
– RRL
Nov 28 '18 at 7:44
$begingroup$
When asking questions here it is important to describe some of your effort in attacking the problem. I'll a partial answer and leave something for you to finish.
$endgroup$
– RRL
Nov 28 '18 at 7:44
$begingroup$
When asking questions here it is important to describe some of your effort in attacking the problem. I'll a partial answer and leave something for you to finish.
$endgroup$
– RRL
Nov 28 '18 at 7:44
add a comment |
1 Answer
1
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$begingroup$
We have for each $x in E$ the convergence of partial sums as $n to infty$ according to
$$sum_{k=1}^n f_k(x) uparrow sum_{k=1}^infty f_k(x) leqslant +infty$$
Why is it true that the sequence of partial sums is non-decreasing and
must always converge to a possibly extended nonnegative real number?
By the monotone convergence theorem,
$$sum_{k=1}^inftyint_E f_k(x) , dx = lim_{n to infty}sum_{k=1}^n int_E f_k(x) , dx = lim_{n to infty} int_E sum_{k=1}^nf_k(x) , dx = int_E sum_{k=1}^infty f_k(x), dx$$
We are given that the series on the LHS converges, and it follows that
$$int_E sum_{k=1}^infty f_k(x), dx < +infty$$
What does this tell you about $ sum_{k=1}^infty f_k(x)$?
answered Nov 28 '18 at 7:41
RRLRRL
50.5k42573
$endgroup$
add a comment |
Your Answer
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1 Answer
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1 Answer
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$begingroup$
We have for each $x in E$ the convergence of partial sums as $n to infty$ according to
$$sum_{k=1}^n f_k(x) uparrow sum_{k=1}^infty f_k(x) leqslant +infty$$
Why is it true that the sequence of partial sums is non-decreasing and
must always converge to a possibly extended nonnegative real number?
By the monotone convergence theorem,
$$sum_{k=1}^inftyint_E f_k(x) , dx = lim_{n to infty}sum_{k=1}^n int_E f_k(x) , dx = lim_{n to infty} int_E sum_{k=1}^nf_k(x) , dx = int_E sum_{k=1}^infty f_k(x), dx$$
We are given that the series on the LHS converges, and it follows that
$$int_E sum_{k=1}^infty f_k(x), dx < +infty$$
What does this tell you about $ sum_{k=1}^infty f_k(x)$?
answered Nov 28 '18 at 7:41
RRLRRL
50.5k42573
$endgroup$
add a comment |
$begingroup$
We have for each $x in E$ the convergence of partial sums as $n to infty$ according to
$$sum_{k=1}^n f_k(x) uparrow sum_{k=1}^infty f_k(x) leqslant +infty$$
Why is it true that the sequence of partial sums is non-decreasing and
must always converge to a possibly extended nonnegative real number?
By the monotone convergence theorem,
$$sum_{k=1}^inftyint_E f_k(x) , dx = lim_{n to infty}sum_{k=1}^n int_E f_k(x) , dx = lim_{n to infty} int_E sum_{k=1}^nf_k(x) , dx = int_E sum_{k=1}^infty f_k(x), dx$$
We are given that the series on the LHS converges, and it follows that
$$int_E sum_{k=1}^infty f_k(x), dx < +infty$$
What does this tell you about $ sum_{k=1}^infty f_k(x)$?
answered Nov 28 '18 at 7:41
RRLRRL
50.5k42573
$endgroup$
add a comment |
$begingroup$
We have for each $x in E$ the convergence of partial sums as $n to infty$ according to
$$sum_{k=1}^n f_k(x) uparrow sum_{k=1}^infty f_k(x) leqslant +infty$$
Why is it true that the sequence of partial sums is non-decreasing and
must always converge to a possibly extended nonnegative real number?
By the monotone convergence theorem,
$$sum_{k=1}^inftyint_E f_k(x) , dx = lim_{n to infty}sum_{k=1}^n int_E f_k(x) , dx = lim_{n to infty} int_E sum_{k=1}^nf_k(x) , dx = int_E sum_{k=1}^infty f_k(x), dx$$
We are given that the series on the LHS converges, and it follows that
$$int_E sum_{k=1}^infty f_k(x), dx < +infty$$
What does this tell you about $ sum_{k=1}^infty f_k(x)$?
answered Nov 28 '18 at 7:41
RRLRRL
50.5k42573
$endgroup$
We have for each $x in E$ the convergence of partial sums as $n to infty$ according to
$$sum_{k=1}^n f_k(x) uparrow sum_{k=1}^infty f_k(x) leqslant +infty$$
Why is it true that the sequence of partial sums is non-decreasing and
must always converge to a possibly extended nonnegative real number?
By the monotone convergence theorem,
$$sum_{k=1}^inftyint_E f_k(x) , dx = lim_{n to infty}sum_{k=1}^n int_E f_k(x) , dx = lim_{n to infty} int_E sum_{k=1}^nf_k(x) , dx = int_E sum_{k=1}^infty f_k(x), dx$$
We are given that the series on the LHS converges, and it follows that
$$int_E sum_{k=1}^infty f_k(x), dx < +infty$$
What does this tell you about $ sum_{k=1}^infty f_k(x)$?
answered Nov 28 '18 at 7:41
RRLRRL
50.5k42573
edited Nov 28 '18 at 7:50
answered Nov 28 '18 at 7:41
RRLRRL
50.5k42573
answered Nov 28 '18 at 7:41
RRLRRL
50.5k42573
answered Nov 28 '18 at 7:41
RRLRRL
50.5k42573
50.5k42573
add a comment |
add a comment |
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$begingroup$
When asking questions here it is important to describe some of your effort in attacking the problem. I'll a partial answer and leave something for you to finish.
$endgroup$
– RRL
Nov 28 '18 at 7:44