Measurable sets in pratice with Lebesgue-Stieltjes measure
$begingroup$
If we consider $F colon mathbb{R} to mathbb{R},$ defined as
$$
F(x) = left{begin{array}{cc} 0, & mbox{if } x < 0 \ 3, & mbox{if } 0 le x < 4 \ 8, &mbox{otherwise.} end{array}right.,
$$
and let $mu_F$ the corresponding Lebesgue-Stieltjes measure. What are the measurable sets in this measure? That is, the sets $E$ satisfying
$$ mu_F(A) = mu_F(A cap E) + mu_F(A setminus E) $$
$forall A in mathbb{R}.$
real-analysis measure-theory lebesgue-measure
asked Dec 3 '18 at 3:08
674123173797 - 4674123173797 - 4
1458
$endgroup$
add a comment |
$begingroup$
If we consider $F colon mathbb{R} to mathbb{R},$ defined as
$$
F(x) = left{begin{array}{cc} 0, & mbox{if } x < 0 \ 3, & mbox{if } 0 le x < 4 \ 8, &mbox{otherwise.} end{array}right.,
$$
and let $mu_F$ the corresponding Lebesgue-Stieltjes measure. What are the measurable sets in this measure? That is, the sets $E$ satisfying
$$ mu_F(A) = mu_F(A cap E) + mu_F(A setminus E) $$
$forall A in mathbb{R}.$
real-analysis measure-theory lebesgue-measure
asked Dec 3 '18 at 3:08
674123173797 - 4674123173797 - 4
1458
$endgroup$
add a comment |
$begingroup$
If we consider $F colon mathbb{R} to mathbb{R},$ defined as
$$
F(x) = left{begin{array}{cc} 0, & mbox{if } x < 0 \ 3, & mbox{if } 0 le x < 4 \ 8, &mbox{otherwise.} end{array}right.,
$$
and let $mu_F$ the corresponding Lebesgue-Stieltjes measure. What are the measurable sets in this measure? That is, the sets $E$ satisfying
$$ mu_F(A) = mu_F(A cap E) + mu_F(A setminus E) $$
$forall A in mathbb{R}.$
real-analysis measure-theory lebesgue-measure
asked Dec 3 '18 at 3:08
674123173797 - 4674123173797 - 4
1458
$endgroup$
If we consider $F colon mathbb{R} to mathbb{R},$ defined as
$$
F(x) = left{begin{array}{cc} 0, & mbox{if } x < 0 \ 3, & mbox{if } 0 le x < 4 \ 8, &mbox{otherwise.} end{array}right.,
$$
and let $mu_F$ the corresponding Lebesgue-Stieltjes measure. What are the measurable sets in this measure? That is, the sets $E$ satisfying
$$ mu_F(A) = mu_F(A cap E) + mu_F(A setminus E) $$
$forall A in mathbb{R}.$
real-analysis measure-theory lebesgue-measure
real-analysis measure-theory lebesgue-measure
asked Dec 3 '18 at 3:08
674123173797 - 4674123173797 - 4
1458
asked Dec 3 '18 at 3:08
674123173797 - 4674123173797 - 4
1458
asked Dec 3 '18 at 3:08
674123173797 - 4674123173797 - 4
1458
asked Dec 3 '18 at 3:08
674123173797 - 4674123173797 - 4
1458
asked Dec 3 '18 at 3:08
674123173797 - 4674123173797 - 4
1458
1458
add a comment |
add a comment |
1 Answer
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$begingroup$
$mu_F(A)=3delta_0(A)+4delta_4(A)$ where $delta_x(A)=1$ if $x in A$ and $0$ otherwise. Since $delta_x(A)= delta_x(Acap E)+delta_x(Asetminus E)$ for all $A,E subset mathbb R$ it follows that all sets are measurable for $mu_F$.
answered Dec 3 '18 at 6:25
Kavi Rama MurthyKavi Rama Murthy
61.3k42262
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1 Answer
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1 Answer
1
active
oldest
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active
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votes
$begingroup$
$mu_F(A)=3delta_0(A)+4delta_4(A)$ where $delta_x(A)=1$ if $x in A$ and $0$ otherwise. Since $delta_x(A)= delta_x(Acap E)+delta_x(Asetminus E)$ for all $A,E subset mathbb R$ it follows that all sets are measurable for $mu_F$.
answered Dec 3 '18 at 6:25
Kavi Rama MurthyKavi Rama Murthy
61.3k42262
$endgroup$
add a comment |
$begingroup$
$mu_F(A)=3delta_0(A)+4delta_4(A)$ where $delta_x(A)=1$ if $x in A$ and $0$ otherwise. Since $delta_x(A)= delta_x(Acap E)+delta_x(Asetminus E)$ for all $A,E subset mathbb R$ it follows that all sets are measurable for $mu_F$.
answered Dec 3 '18 at 6:25
Kavi Rama MurthyKavi Rama Murthy
61.3k42262
$endgroup$
add a comment |
$begingroup$
$mu_F(A)=3delta_0(A)+4delta_4(A)$ where $delta_x(A)=1$ if $x in A$ and $0$ otherwise. Since $delta_x(A)= delta_x(Acap E)+delta_x(Asetminus E)$ for all $A,E subset mathbb R$ it follows that all sets are measurable for $mu_F$.
answered Dec 3 '18 at 6:25
Kavi Rama MurthyKavi Rama Murthy
61.3k42262
$endgroup$
$mu_F(A)=3delta_0(A)+4delta_4(A)$ where $delta_x(A)=1$ if $x in A$ and $0$ otherwise. Since $delta_x(A)= delta_x(Acap E)+delta_x(Asetminus E)$ for all $A,E subset mathbb R$ it follows that all sets are measurable for $mu_F$.
answered Dec 3 '18 at 6:25
Kavi Rama MurthyKavi Rama Murthy
61.3k42262
answered Dec 3 '18 at 6:25
Kavi Rama MurthyKavi Rama Murthy
61.3k42262
answered Dec 3 '18 at 6:25
Kavi Rama MurthyKavi Rama Murthy
61.3k42262
answered Dec 3 '18 at 6:25
Kavi Rama MurthyKavi Rama Murthy
61.3k42262
61.3k42262
add a comment |
add a comment |
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