Timeline for Relation between measure of sets
Current License: CC BY-SA 3.0
17 events
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| Nov 23, 2012 at 17:42 | vote | accept | QuantumLogarithm | ||
| Nov 12, 2012 at 11:09 | history | edited | QuantumLogarithm | CC BY-SA 3.0 |
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| Nov 12, 2012 at 11:00 | history | edited | QuantumLogarithm | CC BY-SA 3.0 |
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| Nov 12, 2012 at 10:56 | comment | added | GH from MO | Lorenzo: If both $A$ and $B$ is a union of $C_\sigma$'s, then $\mu(A\cap C_\sigma)\mu(B\cap C_\sigma)/\mu(C_\sigma)$ equals $\mu(C_\sigma)$ when $C_\sigma\subset A\cap B$ and zero otherwise, hence the right hand side in your inequality is $\mu(A\cap B)/\mu(A)$. That is, your inequality is trivial in this case, and if you omit the $\mu(A)$ from the denominator then it becomes an identity. | |
| Nov 12, 2012 at 10:49 | history | edited | QuantumLogarithm | CC BY-SA 3.0 |
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| Nov 12, 2012 at 10:40 | comment | added | QuantumLogarithm | I have added the contest. In particular I have specified that I would like to show that the inequality is an equality if both A and B are equal to the union of a finite number of sets C_sigma. | |
| Nov 12, 2012 at 10:28 | history | edited | QuantumLogarithm | CC BY-SA 3.0 |
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| Nov 12, 2012 at 10:26 | answer | added | Sergei Ivanov | timeline score: 6 | |
| Nov 12, 2012 at 10:17 | comment | added | anstei | The negative vote is not from me. But I agree with HW that some context might help. | |
| Nov 12, 2012 at 10:15 | comment | added | GH from MO | Lorenzo: The inequality is true for $A=B$. By Cauchy-Schwarz, $\mu(A)^2=(\sum_\sigma\mu(A\cap C_\sigma))^2\leq(\sum_\sigma\mu(C_\sigma))(\sum_\sigma \mu(A\cap C_\sigma)^2/\mu(C_\sigma))$. Rearranging, you get your inequality for $A=B$. | |
| Nov 12, 2012 at 10:08 | comment | added | HJRW | Lorenzo, I didn't case the negative vote, but without any context this question looks like homework. Perhaps you could explain the context in which the question arose? You might also like to look at the FAQ for guidelines on asking questions. | |
| Nov 12, 2012 at 10:07 | history | edited | QuantumLogarithm | CC BY-SA 3.0 |
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| Nov 12, 2012 at 10:07 | comment | added | GH from MO | I think this is a fine question. For $A=B$ it is equivalent to the Cauchy-Schwarz inequality, so it is not trivial (if true). | |
| Nov 12, 2012 at 10:04 | comment | added | QuantumLogarithm | I wrote in that way to let more evident the corrispondence between the terms corresponding to the same sigma... | |
| Nov 12, 2012 at 10:03 | comment | added | QuantumLogarithm | This is obvious, but the question involves the relation with the right term! Take back your negative vote!!! | |
| Nov 12, 2012 at 9:29 | comment | added | anstei | The LHS is simply $\mu(A\cap B)$ by finite additivity. | |
| Nov 12, 2012 at 9:20 | history | asked | QuantumLogarithm | CC BY-SA 3.0 |