Mixing
Measures on $mathbb{R}$ which are invariant under addition and multiplication
0
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Is there a finite measure on $mathbb{R}$ which is invariant both under addition and multiplication?
i.e., is there a finite measure $mu$ s.t. $mu(rA)=mu(A)$ and $mu(r+A)=mu(A)$ for all $rin mathbb{R}$ and measurable $Asubset mathbb{R}$?
real-analysis
asked Jan 30 at 13:23
user17488user17488
889
$endgroup$
add a comment |
0
$begingroup$
Is there a finite measure on $mathbb{R}$ which is invariant both under addition and multiplication?
i.e., is there a finite measure $mu$ s.t. $mu(rA)=mu(A)$ and $mu(r+A)=mu(A)$ for all $rin mathbb{R}$ and measurable $Asubset mathbb{R}$?
real-analysis
asked Jan 30 at 13:23
user17488user17488
889
$endgroup$
$begingroup$
Please specify what you mean by addition and multiplication.
$endgroup$
– Klaus
Jan 30 at 13:25
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The zero measure will do.
$endgroup$
– Will M.
Jan 31 at 7:49
1
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Depends on your sigma-algebra. In the Borel-sigma-algebra there is only the zero measure with those properties. But if you take for example ${ emptyset, mathbb{R} }$ as your sigma algebra, then you have some more.
$endgroup$
– Severin Schraven
Jan 31 at 7:51
add a comment |
0
0
0
$begingroup$
Is there a finite measure on $mathbb{R}$ which is invariant both under addition and multiplication?
i.e., is there a finite measure $mu$ s.t. $mu(rA)=mu(A)$ and $mu(r+A)=mu(A)$ for all $rin mathbb{R}$ and measurable $Asubset mathbb{R}$?
real-analysis
asked Jan 30 at 13:23
user17488user17488
889
$endgroup$
Is there a finite measure on $mathbb{R}$ which is invariant both under addition and multiplication?
i.e., is there a finite measure $mu$ s.t. $mu(rA)=mu(A)$ and $mu(r+A)=mu(A)$ for all $rin mathbb{R}$ and measurable $Asubset mathbb{R}$?
real-analysis
real-analysis
asked Jan 30 at 13:23
user17488user17488
889
asked Jan 30 at 13:23
user17488user17488
889
edited Jan 31 at 7:44
user17488
asked Jan 30 at 13:23
user17488user17488
889
asked Jan 30 at 13:23
user17488user17488
889
asked Jan 30 at 13:23
user17488user17488
889
889
$begingroup$
Please specify what you mean by addition and multiplication.
$endgroup$
– Klaus
Jan 30 at 13:25
$begingroup$
The zero measure will do.
$endgroup$
– Will M.
Jan 31 at 7:49
1
$begingroup$
Depends on your sigma-algebra. In the Borel-sigma-algebra there is only the zero measure with those properties. But if you take for example ${ emptyset, mathbb{R} }$ as your sigma algebra, then you have some more.
$endgroup$
– Severin Schraven
Jan 31 at 7:51
add a comment |
$begingroup$
Please specify what you mean by addition and multiplication.
$endgroup$
– Klaus
Jan 30 at 13:25
$begingroup$
The zero measure will do.
$endgroup$
– Will M.
Jan 31 at 7:49
1
$begingroup$
Depends on your sigma-algebra. In the Borel-sigma-algebra there is only the zero measure with those properties. But if you take for example ${ emptyset, mathbb{R} }$ as your sigma algebra, then you have some more.
$endgroup$
– Severin Schraven
Jan 31 at 7:51
$begingroup$
Please specify what you mean by addition and multiplication.
$endgroup$
– Klaus
Jan 30 at 13:25
$begingroup$
Please specify what you mean by addition and multiplication.
$endgroup$
– Klaus
Jan 30 at 13:25
$begingroup$
The zero measure will do.
$endgroup$
– Will M.
Jan 31 at 7:49
$begingroup$
The zero measure will do.
$endgroup$
– Will M.
Jan 31 at 7:49
1
1
$begingroup$
Depends on your sigma-algebra. In the Borel-sigma-algebra there is only the zero measure with those properties. But if you take for example ${ emptyset, mathbb{R} }$ as your sigma algebra, then you have some more.
$endgroup$
– Severin Schraven
Jan 31 at 7:51
$begingroup$
Depends on your sigma-algebra. In the Borel-sigma-algebra there is only the zero measure with those properties. But if you take for example ${ emptyset, mathbb{R} }$ as your sigma algebra, then you have some more.
$endgroup$
– Severin Schraven
Jan 31 at 7:51
add a comment |
0
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$begingroup$
Please specify what you mean by addition and multiplication.
$endgroup$
– Klaus
Jan 30 at 13:25
$begingroup$
The zero measure will do.
$endgroup$
– Will M.
Jan 31 at 7:49
1
$begingroup$
Depends on your sigma-algebra. In the Borel-sigma-algebra there is only the zero measure with those properties. But if you take for example ${ emptyset, mathbb{R} }$ as your sigma algebra, then you have some more.
$endgroup$
– Severin Schraven
Jan 31 at 7:51