Mixing
Lebesgue measure vs Lebesgue-Stieltjes measure
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Reading advanced probability theory book I've come across Lebesgue-Stieltjes measure. Could someone explain what is the difference between it and "standard" Lebesgue measure on $mathbb{R}$? Thank you.
probability-theory measure-theory
asked Nov 16 '15 at 20:41
luka5zluka5z
2,23021647
$endgroup$
add a comment |
$begingroup$
Reading advanced probability theory book I've come across Lebesgue-Stieltjes measure. Could someone explain what is the difference between it and "standard" Lebesgue measure on $mathbb{R}$? Thank you.
probability-theory measure-theory
asked Nov 16 '15 at 20:41
luka5zluka5z
2,23021647
$endgroup$
$begingroup$
Lebesgue integral: $int f(x);dx$. Lebesgue-Stieltjes integral: $int f(x);dg(x)$.
$endgroup$
– GEdgar
Jan 8 at 1:09
add a comment |
$begingroup$
Reading advanced probability theory book I've come across Lebesgue-Stieltjes measure. Could someone explain what is the difference between it and "standard" Lebesgue measure on $mathbb{R}$? Thank you.
probability-theory measure-theory
asked Nov 16 '15 at 20:41
luka5zluka5z
2,23021647
$endgroup$
Reading advanced probability theory book I've come across Lebesgue-Stieltjes measure. Could someone explain what is the difference between it and "standard" Lebesgue measure on $mathbb{R}$? Thank you.
probability-theory measure-theory
probability-theory measure-theory
asked Nov 16 '15 at 20:41
luka5zluka5z
2,23021647
asked Nov 16 '15 at 20:41
luka5zluka5z
2,23021647
asked Nov 16 '15 at 20:41
luka5zluka5z
2,23021647
asked Nov 16 '15 at 20:41
luka5zluka5z
2,23021647
asked Nov 16 '15 at 20:41
luka5zluka5z
2,23021647
2,23021647
$begingroup$
Lebesgue integral: $int f(x);dx$. Lebesgue-Stieltjes integral: $int f(x);dg(x)$.
$endgroup$
– GEdgar
Jan 8 at 1:09
add a comment |
$begingroup$
Lebesgue integral: $int f(x);dx$. Lebesgue-Stieltjes integral: $int f(x);dg(x)$.
$endgroup$
– GEdgar
Jan 8 at 1:09
$begingroup$
Lebesgue integral: $int f(x);dx$. Lebesgue-Stieltjes integral: $int f(x);dg(x)$.
$endgroup$
– GEdgar
Jan 8 at 1:09
$begingroup$
Lebesgue integral: $int f(x);dx$. Lebesgue-Stieltjes integral: $int f(x);dg(x)$.
$endgroup$
– GEdgar
Jan 8 at 1:09
add a comment |
1 Answer
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It can be proven that for any increasing right-contineous function $F:mathbb{R}rightarrow mathbb{R}$ there exist one and only one measure $mu_F$ on $(mathbb{R},mathcal{B}(mathbb{R}))$ satisfying the property:
$$ mu_F((a,b])=F(b)-F(a) $$
for every interval (a,b] with $a<b$.
$mu_F$ is called the Lebesgue-Stieltjes measure belonging to F. The Lebesgue measure is simply the Lebesgue-Stieltjes measure belonging to the identity function.
answered Jan 7 at 22:25
Leander Tilsted KristensenLeander Tilsted Kristensen
213
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1 Answer
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1 Answer
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$begingroup$
It can be proven that for any increasing right-contineous function $F:mathbb{R}rightarrow mathbb{R}$ there exist one and only one measure $mu_F$ on $(mathbb{R},mathcal{B}(mathbb{R}))$ satisfying the property:
$$ mu_F((a,b])=F(b)-F(a) $$
for every interval (a,b] with $a<b$.
$mu_F$ is called the Lebesgue-Stieltjes measure belonging to F. The Lebesgue measure is simply the Lebesgue-Stieltjes measure belonging to the identity function.
answered Jan 7 at 22:25
Leander Tilsted KristensenLeander Tilsted Kristensen
213
$endgroup$
add a comment |
$begingroup$
It can be proven that for any increasing right-contineous function $F:mathbb{R}rightarrow mathbb{R}$ there exist one and only one measure $mu_F$ on $(mathbb{R},mathcal{B}(mathbb{R}))$ satisfying the property:
$$ mu_F((a,b])=F(b)-F(a) $$
for every interval (a,b] with $a<b$.
$mu_F$ is called the Lebesgue-Stieltjes measure belonging to F. The Lebesgue measure is simply the Lebesgue-Stieltjes measure belonging to the identity function.
answered Jan 7 at 22:25
Leander Tilsted KristensenLeander Tilsted Kristensen
213
$endgroup$
add a comment |
$begingroup$
It can be proven that for any increasing right-contineous function $F:mathbb{R}rightarrow mathbb{R}$ there exist one and only one measure $mu_F$ on $(mathbb{R},mathcal{B}(mathbb{R}))$ satisfying the property:
$$ mu_F((a,b])=F(b)-F(a) $$
for every interval (a,b] with $a<b$.
$mu_F$ is called the Lebesgue-Stieltjes measure belonging to F. The Lebesgue measure is simply the Lebesgue-Stieltjes measure belonging to the identity function.
answered Jan 7 at 22:25
Leander Tilsted KristensenLeander Tilsted Kristensen
213
$endgroup$
It can be proven that for any increasing right-contineous function $F:mathbb{R}rightarrow mathbb{R}$ there exist one and only one measure $mu_F$ on $(mathbb{R},mathcal{B}(mathbb{R}))$ satisfying the property:
$$ mu_F((a,b])=F(b)-F(a) $$
for every interval (a,b] with $a<b$.
$mu_F$ is called the Lebesgue-Stieltjes measure belonging to F. The Lebesgue measure is simply the Lebesgue-Stieltjes measure belonging to the identity function.
answered Jan 7 at 22:25
Leander Tilsted KristensenLeander Tilsted Kristensen
213
answered Jan 7 at 22:25
Leander Tilsted KristensenLeander Tilsted Kristensen
213
answered Jan 7 at 22:25
Leander Tilsted KristensenLeander Tilsted Kristensen
213
answered Jan 7 at 22:25
Leander Tilsted KristensenLeander Tilsted Kristensen
213
213
add a comment |
add a comment |
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$begingroup$
Lebesgue integral: $int f(x);dx$. Lebesgue-Stieltjes integral: $int f(x);dg(x)$.
$endgroup$
– GEdgar
Jan 8 at 1:09