Mixing
If either $f_n longrightarrow f $ almost everywhere OR $f_n longrightarrow f$
[1] Are all simple functions Lebesgue Integrable? Prove or disprove.
[2] If either $f_n longrightarrow f $ almost everywhere OR $f_n longrightarrow f$ in measure then show that f is finite valued (ie real valued) almost everywhere.
MY ATTEMPT
[1] I'm not sure if all simple functions are Lebesgue integrable or not but this is my example of a simple function which is not Lebesgue integrable
Let $(mathbb{R},B_mathbb{R},mu_L)$ be a Lebesgue measure space and
$phi:[0,infty] longrightarrow mathbb{R}$ is a simple function defined as
$$
phi(x)=
begin{cases}
-1, if x in bigcup_{x in mathbb{Z^+}} [2k+1,2k+2],\
1, if x in bigcup_{x in mathbb{Z^+}} [2k,2k+1]
end{cases}
$$
[2] Not sure how to start or which one to pick.In either case i'm not sure if I have to take some sub sequence and then show it is finite and if so what's my logic and steps?
measure-theory convergence almost-everywhere measurable-functions
asked Nov 21 '18 at 23:25
Jason MooreJason Moore
607
add a comment |
[1] Are all simple functions Lebesgue Integrable? Prove or disprove.
[2] If either $f_n longrightarrow f $ almost everywhere OR $f_n longrightarrow f$ in measure then show that f is finite valued (ie real valued) almost everywhere.
MY ATTEMPT
[1] I'm not sure if all simple functions are Lebesgue integrable or not but this is my example of a simple function which is not Lebesgue integrable
Let $(mathbb{R},B_mathbb{R},mu_L)$ be a Lebesgue measure space and
$phi:[0,infty] longrightarrow mathbb{R}$ is a simple function defined as
$$
phi(x)=
begin{cases}
-1, if x in bigcup_{x in mathbb{Z^+}} [2k+1,2k+2],\
1, if x in bigcup_{x in mathbb{Z^+}} [2k,2k+1]
end{cases}
$$
[2] Not sure how to start or which one to pick.In either case i'm not sure if I have to take some sub sequence and then show it is finite and if so what's my logic and steps?
measure-theory convergence almost-everywhere measurable-functions
asked Nov 21 '18 at 23:25
Jason MooreJason Moore
607
add a comment |
[1] Are all simple functions Lebesgue Integrable? Prove or disprove.
[2] If either $f_n longrightarrow f $ almost everywhere OR $f_n longrightarrow f$ in measure then show that f is finite valued (ie real valued) almost everywhere.
MY ATTEMPT
[1] I'm not sure if all simple functions are Lebesgue integrable or not but this is my example of a simple function which is not Lebesgue integrable
Let $(mathbb{R},B_mathbb{R},mu_L)$ be a Lebesgue measure space and
$phi:[0,infty] longrightarrow mathbb{R}$ is a simple function defined as
$$
phi(x)=
begin{cases}
-1, if x in bigcup_{x in mathbb{Z^+}} [2k+1,2k+2],\
1, if x in bigcup_{x in mathbb{Z^+}} [2k,2k+1]
end{cases}
$$
[2] Not sure how to start or which one to pick.In either case i'm not sure if I have to take some sub sequence and then show it is finite and if so what's my logic and steps?
measure-theory convergence almost-everywhere measurable-functions
asked Nov 21 '18 at 23:25
Jason MooreJason Moore
607
[1] Are all simple functions Lebesgue Integrable? Prove or disprove.
[2] If either $f_n longrightarrow f $ almost everywhere OR $f_n longrightarrow f$ in measure then show that f is finite valued (ie real valued) almost everywhere.
MY ATTEMPT
[1] I'm not sure if all simple functions are Lebesgue integrable or not but this is my example of a simple function which is not Lebesgue integrable
Let $(mathbb{R},B_mathbb{R},mu_L)$ be a Lebesgue measure space and
$phi:[0,infty] longrightarrow mathbb{R}$ is a simple function defined as
$$
phi(x)=
begin{cases}
-1, if x in bigcup_{x in mathbb{Z^+}} [2k+1,2k+2],\
1, if x in bigcup_{x in mathbb{Z^+}} [2k,2k+1]
end{cases}
$$
[2] Not sure how to start or which one to pick.In either case i'm not sure if I have to take some sub sequence and then show it is finite and if so what's my logic and steps?
measure-theory convergence almost-everywhere measurable-functions
measure-theory convergence almost-everywhere measurable-functions
asked Nov 21 '18 at 23:25
Jason MooreJason Moore
607
asked Nov 21 '18 at 23:25
Jason MooreJason Moore
607
asked Nov 21 '18 at 23:25
Jason MooreJason Moore
607
asked Nov 21 '18 at 23:25
Jason MooreJason Moore
607
asked Nov 21 '18 at 23:25
Jason MooreJason Moore
607
607
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
[1] Think of a constant.
[2] For the a.e. convergence, remember a convergent sequence is bounded. For the convergence in measure remember there is a subsequence converging a.e.
answered Nov 21 '18 at 23:31
Will M.Will M.
2,442314
Ok I get one but I'm still pretty lost in 2 mate. Anything else you can tell me?
– Jason Moore
Nov 22 '18 at 0:40
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
[1] Think of a constant.
[2] For the a.e. convergence, remember a convergent sequence is bounded. For the convergence in measure remember there is a subsequence converging a.e.
answered Nov 21 '18 at 23:31
Will M.Will M.
2,442314
Ok I get one but I'm still pretty lost in 2 mate. Anything else you can tell me?
– Jason Moore
Nov 22 '18 at 0:40
add a comment |
[1] Think of a constant.
[2] For the a.e. convergence, remember a convergent sequence is bounded. For the convergence in measure remember there is a subsequence converging a.e.
answered Nov 21 '18 at 23:31
Will M.Will M.
2,442314
Ok I get one but I'm still pretty lost in 2 mate. Anything else you can tell me?
– Jason Moore
Nov 22 '18 at 0:40
add a comment |
[1] Think of a constant.
[2] For the a.e. convergence, remember a convergent sequence is bounded. For the convergence in measure remember there is a subsequence converging a.e.
answered Nov 21 '18 at 23:31
Will M.Will M.
2,442314
[1] Think of a constant.
[2] For the a.e. convergence, remember a convergent sequence is bounded. For the convergence in measure remember there is a subsequence converging a.e.
answered Nov 21 '18 at 23:31
Will M.Will M.
2,442314
answered Nov 21 '18 at 23:31
Will M.Will M.
2,442314
answered Nov 21 '18 at 23:31
Will M.Will M.
2,442314
answered Nov 21 '18 at 23:31
Will M.Will M.
2,442314
2,442314
Ok I get one but I'm still pretty lost in 2 mate. Anything else you can tell me?
– Jason Moore
Nov 22 '18 at 0:40
add a comment |
Ok I get one but I'm still pretty lost in 2 mate. Anything else you can tell me?
– Jason Moore
Nov 22 '18 at 0:40
Ok I get one but I'm still pretty lost in 2 mate. Anything else you can tell me?
– Jason Moore
Nov 22 '18 at 0:40
Ok I get one but I'm still pretty lost in 2 mate. Anything else you can tell me?
– Jason Moore
Nov 22 '18 at 0:40
add a comment |
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