What does it mean for a function $f$ to be continuous almost every where with respect to the measure induced...
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In this case $g$ is monotone increasing on the finite interval $[a,b]$. Does this mean that $f$ is continuous in the normal sense of continuity (ie. $|f(x)-f(y)| < epsilon$ when $|x-y|<delta$) except on a set of measure $0$ where the measure here is the measure induced by $g$?
For clarification here is he whole question I am asked to answer.
Let $g$ be a monotone function on a finite closed interval $[a,b]$. Show that a bounded function $f$ defined on $[a,b]$ is Riemann-Stieltjes integrable with respect to $g$ iff $f$ is continuous almost everywhere with respect to the measure induced by $g$.
Im guessing the measure induced by $g$ is the Borel measure, $mu_g$, with the property $mu_g(a,b) = g(b)-g(a)$ since $g$ is monotone.
measure-theory continuity
asked Dec 7 '18 at 18:22
alpastalpast
468314
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show 6 more comments
$begingroup$
In this case $g$ is monotone increasing on the finite interval $[a,b]$. Does this mean that $f$ is continuous in the normal sense of continuity (ie. $|f(x)-f(y)| < epsilon$ when $|x-y|<delta$) except on a set of measure $0$ where the measure here is the measure induced by $g$?
For clarification here is he whole question I am asked to answer.
Let $g$ be a monotone function on a finite closed interval $[a,b]$. Show that a bounded function $f$ defined on $[a,b]$ is Riemann-Stieltjes integrable with respect to $g$ iff $f$ is continuous almost everywhere with respect to the measure induced by $g$.
Im guessing the measure induced by $g$ is the Borel measure, $mu_g$, with the property $mu_g(a,b) = g(b)-g(a)$ since $g$ is monotone.
measure-theory continuity
asked Dec 7 '18 at 18:22
alpastalpast
468314
$endgroup$
$begingroup$
"the measure induced by $g$": you mean $g(x)dx$ or $dg(x)$?
$endgroup$
– Federico
Dec 7 '18 at 18:24
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Riemann-Stieltjes measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:25
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@Federico what is the Riemann-Stiltjies measure induced by $g$?
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– mathworker21
Dec 7 '18 at 18:25
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What is the measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:26
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In any case, the answer would be yes; regardless of what measure you are considering
$endgroup$
– Federico
Dec 7 '18 at 18:26
|
show 6 more comments
$begingroup$
In this case $g$ is monotone increasing on the finite interval $[a,b]$. Does this mean that $f$ is continuous in the normal sense of continuity (ie. $|f(x)-f(y)| < epsilon$ when $|x-y|<delta$) except on a set of measure $0$ where the measure here is the measure induced by $g$?
For clarification here is he whole question I am asked to answer.
Let $g$ be a monotone function on a finite closed interval $[a,b]$. Show that a bounded function $f$ defined on $[a,b]$ is Riemann-Stieltjes integrable with respect to $g$ iff $f$ is continuous almost everywhere with respect to the measure induced by $g$.
Im guessing the measure induced by $g$ is the Borel measure, $mu_g$, with the property $mu_g(a,b) = g(b)-g(a)$ since $g$ is monotone.
measure-theory continuity
asked Dec 7 '18 at 18:22
alpastalpast
468314
$endgroup$
In this case $g$ is monotone increasing on the finite interval $[a,b]$. Does this mean that $f$ is continuous in the normal sense of continuity (ie. $|f(x)-f(y)| < epsilon$ when $|x-y|<delta$) except on a set of measure $0$ where the measure here is the measure induced by $g$?
For clarification here is he whole question I am asked to answer.
Let $g$ be a monotone function on a finite closed interval $[a,b]$. Show that a bounded function $f$ defined on $[a,b]$ is Riemann-Stieltjes integrable with respect to $g$ iff $f$ is continuous almost everywhere with respect to the measure induced by $g$.
Im guessing the measure induced by $g$ is the Borel measure, $mu_g$, with the property $mu_g(a,b) = g(b)-g(a)$ since $g$ is monotone.
measure-theory continuity
measure-theory continuity
asked Dec 7 '18 at 18:22
alpastalpast
468314
asked Dec 7 '18 at 18:22
alpastalpast
468314
edited Dec 7 '18 at 18:29
alpast
asked Dec 7 '18 at 18:22
alpastalpast
468314
asked Dec 7 '18 at 18:22
alpastalpast
468314
asked Dec 7 '18 at 18:22
alpastalpast
468314
468314
$begingroup$
"the measure induced by $g$": you mean $g(x)dx$ or $dg(x)$?
$endgroup$
– Federico
Dec 7 '18 at 18:24
$begingroup$
Riemann-Stieltjes measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:25
$begingroup$
@Federico what is the Riemann-Stiltjies measure induced by $g$?
$endgroup$
– mathworker21
Dec 7 '18 at 18:25
$begingroup$
What is the measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:26
$begingroup$
In any case, the answer would be yes; regardless of what measure you are considering
$endgroup$
– Federico
Dec 7 '18 at 18:26
|
show 6 more comments
$begingroup$
"the measure induced by $g$": you mean $g(x)dx$ or $dg(x)$?
$endgroup$
– Federico
Dec 7 '18 at 18:24
$begingroup$
Riemann-Stieltjes measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:25
$begingroup$
@Federico what is the Riemann-Stiltjies measure induced by $g$?
$endgroup$
– mathworker21
Dec 7 '18 at 18:25
$begingroup$
What is the measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:26
$begingroup$
In any case, the answer would be yes; regardless of what measure you are considering
$endgroup$
– Federico
Dec 7 '18 at 18:26
$begingroup$
"the measure induced by $g$": you mean $g(x)dx$ or $dg(x)$?
$endgroup$
– Federico
Dec 7 '18 at 18:24
$begingroup$
"the measure induced by $g$": you mean $g(x)dx$ or $dg(x)$?
$endgroup$
– Federico
Dec 7 '18 at 18:24
$begingroup$
Riemann-Stieltjes measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:25
$begingroup$
Riemann-Stieltjes measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:25
$begingroup$
@Federico what is the Riemann-Stiltjies measure induced by $g$?
$endgroup$
– mathworker21
Dec 7 '18 at 18:25
$begingroup$
@Federico what is the Riemann-Stiltjies measure induced by $g$?
$endgroup$
– mathworker21
Dec 7 '18 at 18:25
$begingroup$
What is the measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:26
$begingroup$
What is the measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:26
$begingroup$
In any case, the answer would be yes; regardless of what measure you are considering
$endgroup$
– Federico
Dec 7 '18 at 18:26
$begingroup$
In any case, the answer would be yes; regardless of what measure you are considering
$endgroup$
– Federico
Dec 7 '18 at 18:26
|
show 6 more comments
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$begingroup$
"the measure induced by $g$": you mean $g(x)dx$ or $dg(x)$?
$endgroup$
– Federico
Dec 7 '18 at 18:24
$begingroup$
Riemann-Stieltjes measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:25
$begingroup$
@Federico what is the Riemann-Stiltjies measure induced by $g$?
$endgroup$
– mathworker21
Dec 7 '18 at 18:25
$begingroup$
What is the measure induced by $g$?
$endgroup$
– Federico
Dec 7 '18 at 18:26
$begingroup$
In any case, the answer would be yes; regardless of what measure you are considering
$endgroup$
– Federico
Dec 7 '18 at 18:26