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Proving existence of dense subset $D subseteq mathbb R$, so that ${f geq alpha} in mathcal{A}$, $forall alpha...
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Background to the question is that I have to show the equivalance of the following three statements:
$(X, mathcal{A})$ a measure space and $f: X to overline{mathbb R}$
Show the equivalence of the three following statements:
$i)$ $f$ is $mathcal{A}-mathcal{B}(overline{mathbb R})-$measureable
$ii)$There exists a dense subset $D subseteq mathbb R$, so that ${f geq alpha}in mathcal{A}$, $forall alpha in D$.
$iii)$ $f^{-1}({+infty}) in mathcal{A}$ and $f^{-1}(mathcal{B}(mathbb R)) subseteq mathcal{A}$
I thought I took the "easy route" when I showed $ii) Rightarrow i)$ and then $i) Rightarrow iii)$. Now I am having trouble proving that $iii) Rightarrow ii)$, which leads me to think that it may not be possible directly and so I'll have to redo my other implications. Is there a way to directly solve:
If $f^{-1}({+infty}) in mathcal{A}$ and $f^{-1}(mathcal{B}(mathbb R)) subseteq mathcal{A} Rightarrow$ I can find $D subseteq mathbb R$ such that ${f geq alpha} in mathcal{A}$, $forall alpha in D$.
real-analysis measure-theory borel-sets
asked Nov 18 at 18:55
SABOY
501211
add a comment |
up vote
0
down vote
favorite
Background to the question is that I have to show the equivalance of the following three statements:
$(X, mathcal{A})$ a measure space and $f: X to overline{mathbb R}$
Show the equivalence of the three following statements:
$i)$ $f$ is $mathcal{A}-mathcal{B}(overline{mathbb R})-$measureable
$ii)$There exists a dense subset $D subseteq mathbb R$, so that ${f geq alpha}in mathcal{A}$, $forall alpha in D$.
$iii)$ $f^{-1}({+infty}) in mathcal{A}$ and $f^{-1}(mathcal{B}(mathbb R)) subseteq mathcal{A}$
I thought I took the "easy route" when I showed $ii) Rightarrow i)$ and then $i) Rightarrow iii)$. Now I am having trouble proving that $iii) Rightarrow ii)$, which leads me to think that it may not be possible directly and so I'll have to redo my other implications. Is there a way to directly solve:
If $f^{-1}({+infty}) in mathcal{A}$ and $f^{-1}(mathcal{B}(mathbb R)) subseteq mathcal{A} Rightarrow$ I can find $D subseteq mathbb R$ such that ${f geq alpha} in mathcal{A}$, $forall alpha in D$.
real-analysis measure-theory borel-sets
asked Nov 18 at 18:55
SABOY
501211
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Background to the question is that I have to show the equivalance of the following three statements:
$(X, mathcal{A})$ a measure space and $f: X to overline{mathbb R}$
Show the equivalence of the three following statements:
$i)$ $f$ is $mathcal{A}-mathcal{B}(overline{mathbb R})-$measureable
$ii)$There exists a dense subset $D subseteq mathbb R$, so that ${f geq alpha}in mathcal{A}$, $forall alpha in D$.
$iii)$ $f^{-1}({+infty}) in mathcal{A}$ and $f^{-1}(mathcal{B}(mathbb R)) subseteq mathcal{A}$
I thought I took the "easy route" when I showed $ii) Rightarrow i)$ and then $i) Rightarrow iii)$. Now I am having trouble proving that $iii) Rightarrow ii)$, which leads me to think that it may not be possible directly and so I'll have to redo my other implications. Is there a way to directly solve:
If $f^{-1}({+infty}) in mathcal{A}$ and $f^{-1}(mathcal{B}(mathbb R)) subseteq mathcal{A} Rightarrow$ I can find $D subseteq mathbb R$ such that ${f geq alpha} in mathcal{A}$, $forall alpha in D$.
real-analysis measure-theory borel-sets
asked Nov 18 at 18:55
SABOY
501211
Background to the question is that I have to show the equivalance of the following three statements:
$(X, mathcal{A})$ a measure space and $f: X to overline{mathbb R}$
Show the equivalence of the three following statements:
$i)$ $f$ is $mathcal{A}-mathcal{B}(overline{mathbb R})-$measureable
$ii)$There exists a dense subset $D subseteq mathbb R$, so that ${f geq alpha}in mathcal{A}$, $forall alpha in D$.
$iii)$ $f^{-1}({+infty}) in mathcal{A}$ and $f^{-1}(mathcal{B}(mathbb R)) subseteq mathcal{A}$
I thought I took the "easy route" when I showed $ii) Rightarrow i)$ and then $i) Rightarrow iii)$. Now I am having trouble proving that $iii) Rightarrow ii)$, which leads me to think that it may not be possible directly and so I'll have to redo my other implications. Is there a way to directly solve:
If $f^{-1}({+infty}) in mathcal{A}$ and $f^{-1}(mathcal{B}(mathbb R)) subseteq mathcal{A} Rightarrow$ I can find $D subseteq mathbb R$ such that ${f geq alpha} in mathcal{A}$, $forall alpha in D$.
real-analysis measure-theory borel-sets
real-analysis measure-theory borel-sets
asked Nov 18 at 18:55
SABOY
501211
asked Nov 18 at 18:55
SABOY
501211
edited 2 days ago
asked Nov 18 at 18:55
SABOY
501211
asked Nov 18 at 18:55
SABOY
501211
asked Nov 18 at 18:55
SABOY
501211
501211
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Hint: $f^{-1}([alpha,infty]) = f^{-1}([alpha,infty))cup f^{-1}({infty}).$
answered 2 days ago
zhw.
70.3k43075
How can I use that to find my dense subset $D$?
– SABOY
2 days ago
1
Your dense subset is all of $mathbb R.$
– zhw.
2 days ago
got it, thanks!
– SABOY
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Hint: $f^{-1}([alpha,infty]) = f^{-1}([alpha,infty))cup f^{-1}({infty}).$
answered 2 days ago
zhw.
70.3k43075
How can I use that to find my dense subset $D$?
– SABOY
2 days ago
1
Your dense subset is all of $mathbb R.$
– zhw.
2 days ago
got it, thanks!
– SABOY
2 days ago
add a comment |
up vote
1
down vote
accepted
Hint: $f^{-1}([alpha,infty]) = f^{-1}([alpha,infty))cup f^{-1}({infty}).$
answered 2 days ago
zhw.
70.3k43075
How can I use that to find my dense subset $D$?
– SABOY
2 days ago
1
Your dense subset is all of $mathbb R.$
– zhw.
2 days ago
got it, thanks!
– SABOY
2 days ago
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hint: $f^{-1}([alpha,infty]) = f^{-1}([alpha,infty))cup f^{-1}({infty}).$
answered 2 days ago
zhw.
70.3k43075
Hint: $f^{-1}([alpha,infty]) = f^{-1}([alpha,infty))cup f^{-1}({infty}).$
answered 2 days ago
zhw.
70.3k43075
answered 2 days ago
zhw.
70.3k43075
answered 2 days ago
zhw.
70.3k43075
answered 2 days ago
zhw.
70.3k43075
70.3k43075
How can I use that to find my dense subset $D$?
– SABOY
2 days ago
1
Your dense subset is all of $mathbb R.$
– zhw.
2 days ago
got it, thanks!
– SABOY
2 days ago
add a comment |
How can I use that to find my dense subset $D$?
– SABOY
2 days ago
1
Your dense subset is all of $mathbb R.$
– zhw.
2 days ago
got it, thanks!
– SABOY
2 days ago
How can I use that to find my dense subset $D$?
– SABOY
2 days ago
How can I use that to find my dense subset $D$?
– SABOY
2 days ago
1
1
Your dense subset is all of $mathbb R.$
– zhw.
2 days ago
Your dense subset is all of $mathbb R.$
– zhw.
2 days ago
got it, thanks!
– SABOY
2 days ago
got it, thanks!
– SABOY
2 days ago
add a comment |
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