property of singular measure with respect to Lebesgue measure

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Question: Let $mu$ be a finite positive Borel measure on $mathbbR$ that is singular to Lebesgue measure. Show that
$$lim_rto 0^+ fracmu([x-r,x+r])2r=+infty$$
for $mu$-almost every $xinmathbbR$.



By Folland Theorem 3.22, I know that the set of these $x$ is $m$-null, but I don't know why it is $mu$-a.e.




real-analysis lebesgue-measure singular-measures




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    This is a theorem in Rudin's RCA. Look at the chapter on differentiation of measures.
    – Kavi Rama Murthy
    Jul 26 at 23:39










  • Thank you very much! I just read Folland but not Rudin, so this problem is quite hard for me.
    – QUAN CHEN
    Jul 27 at 5:38














up vote
3
down vote

favorite
1












Question: Let $mu$ be a finite positive Borel measure on $mathbbR$ that is singular to Lebesgue measure. Show that
$$lim_rto 0^+ fracmu([x-r,x+r])2r=+infty$$
for $mu$-almost every $xinmathbbR$.



By Folland Theorem 3.22, I know that the set of these $x$ is $m$-null, but I don't know why it is $mu$-a.e.




real-analysis lebesgue-measure singular-measures




share|cite|improve this question















  • 1




    This is a theorem in Rudin's RCA. Look at the chapter on differentiation of measures.
    – Kavi Rama Murthy
    Jul 26 at 23:39










  • Thank you very much! I just read Folland but not Rudin, so this problem is quite hard for me.
    – QUAN CHEN
    Jul 27 at 5:38












up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





Question: Let $mu$ be a finite positive Borel measure on $mathbbR$ that is singular to Lebesgue measure. Show that
$$lim_rto 0^+ fracmu([x-r,x+r])2r=+infty$$
for $mu$-almost every $xinmathbbR$.



By Folland Theorem 3.22, I know that the set of these $x$ is $m$-null, but I don't know why it is $mu$-a.e.




real-analysis lebesgue-measure singular-measures




share|cite|improve this question











Question: Let $mu$ be a finite positive Borel measure on $mathbbR$ that is singular to Lebesgue measure. Show that
$$lim_rto 0^+ fracmu([x-r,x+r])2r=+infty$$
for $mu$-almost every $xinmathbbR$.



By Folland Theorem 3.22, I know that the set of these $x$ is $m$-null, but I don't know why it is $mu$-a.e.





real-analysis lebesgue-measure singular-measures





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 26 at 22:00









QUAN CHEN

835




835







  • 1




    This is a theorem in Rudin's RCA. Look at the chapter on differentiation of measures.
    – Kavi Rama Murthy
    Jul 26 at 23:39










  • Thank you very much! I just read Folland but not Rudin, so this problem is quite hard for me.
    – QUAN CHEN
    Jul 27 at 5:38












  • 1




    This is a theorem in Rudin's RCA. Look at the chapter on differentiation of measures.
    – Kavi Rama Murthy
    Jul 26 at 23:39










  • Thank you very much! I just read Folland but not Rudin, so this problem is quite hard for me.
    – QUAN CHEN
    Jul 27 at 5:38







1




1




This is a theorem in Rudin's RCA. Look at the chapter on differentiation of measures.
– Kavi Rama Murthy
Jul 26 at 23:39




This is a theorem in Rudin's RCA. Look at the chapter on differentiation of measures.
– Kavi Rama Murthy
Jul 26 at 23:39












Thank you very much! I just read Folland but not Rudin, so this problem is quite hard for me.
– QUAN CHEN
Jul 27 at 5:38




Thank you very much! I just read Folland but not Rudin, so this problem is quite hard for me.
– QUAN CHEN
Jul 27 at 5:38















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