Riemann-Stieltjes Integral - Changing Order of Integration
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I am new to Riemann-Stieltjes integral. I want to ask a very basic question regarding changing the order of integration.
Let $ t > 0 $ and I have an integral that looks like this
$$ int_mathbb{R} int_0^t f(g(x)) dx dg(x). $$
What is the condition so that I change the order of integration? Or mathematically we could write the integral like this
$$ int_mathbb{R} int_0^t f(g(x)) dx dg(x) = int_0^t int_mathbb{R} f(g(x))dg(x)dx $$
riemann-integration
asked Dec 7 '18 at 1:10
BenBen
526
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add a comment |
$begingroup$
I am new to Riemann-Stieltjes integral. I want to ask a very basic question regarding changing the order of integration.
Let $ t > 0 $ and I have an integral that looks like this
$$ int_mathbb{R} int_0^t f(g(x)) dx dg(x). $$
What is the condition so that I change the order of integration? Or mathematically we could write the integral like this
$$ int_mathbb{R} int_0^t f(g(x)) dx dg(x) = int_0^t int_mathbb{R} f(g(x))dg(x)dx $$
riemann-integration
asked Dec 7 '18 at 1:10
BenBen
526
$endgroup$
$begingroup$
I believe we still need the same assumptions Fubini requires. Is $f$ a functoom of a single variable?
$endgroup$
– Sean Roberson
Dec 7 '18 at 4:44
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Function $ f $ is a function of $ g(x) $ and $ x $. For example, the simplest one is $ f(x) = g(x) + x $.
$endgroup$
– Ben
Dec 7 '18 at 16:23
add a comment |
$begingroup$
I am new to Riemann-Stieltjes integral. I want to ask a very basic question regarding changing the order of integration.
Let $ t > 0 $ and I have an integral that looks like this
$$ int_mathbb{R} int_0^t f(g(x)) dx dg(x). $$
What is the condition so that I change the order of integration? Or mathematically we could write the integral like this
$$ int_mathbb{R} int_0^t f(g(x)) dx dg(x) = int_0^t int_mathbb{R} f(g(x))dg(x)dx $$
riemann-integration
asked Dec 7 '18 at 1:10
BenBen
526
$endgroup$
I am new to Riemann-Stieltjes integral. I want to ask a very basic question regarding changing the order of integration.
Let $ t > 0 $ and I have an integral that looks like this
$$ int_mathbb{R} int_0^t f(g(x)) dx dg(x). $$
What is the condition so that I change the order of integration? Or mathematically we could write the integral like this
$$ int_mathbb{R} int_0^t f(g(x)) dx dg(x) = int_0^t int_mathbb{R} f(g(x))dg(x)dx $$
riemann-integration
riemann-integration
asked Dec 7 '18 at 1:10
BenBen
526
asked Dec 7 '18 at 1:10
BenBen
526
asked Dec 7 '18 at 1:10
BenBen
526
asked Dec 7 '18 at 1:10
BenBen
526
asked Dec 7 '18 at 1:10
BenBen
526
526
$begingroup$
I believe we still need the same assumptions Fubini requires. Is $f$ a functoom of a single variable?
$endgroup$
– Sean Roberson
Dec 7 '18 at 4:44
$begingroup$
Function $ f $ is a function of $ g(x) $ and $ x $. For example, the simplest one is $ f(x) = g(x) + x $.
$endgroup$
– Ben
Dec 7 '18 at 16:23
add a comment |
$begingroup$
I believe we still need the same assumptions Fubini requires. Is $f$ a functoom of a single variable?
$endgroup$
– Sean Roberson
Dec 7 '18 at 4:44
$begingroup$
Function $ f $ is a function of $ g(x) $ and $ x $. For example, the simplest one is $ f(x) = g(x) + x $.
$endgroup$
– Ben
Dec 7 '18 at 16:23
$begingroup$
I believe we still need the same assumptions Fubini requires. Is $f$ a functoom of a single variable?
$endgroup$
– Sean Roberson
Dec 7 '18 at 4:44
$begingroup$
I believe we still need the same assumptions Fubini requires. Is $f$ a functoom of a single variable?
$endgroup$
– Sean Roberson
Dec 7 '18 at 4:44
$begingroup$
Function $ f $ is a function of $ g(x) $ and $ x $. For example, the simplest one is $ f(x) = g(x) + x $.
$endgroup$
– Ben
Dec 7 '18 at 16:23
$begingroup$
Function $ f $ is a function of $ g(x) $ and $ x $. For example, the simplest one is $ f(x) = g(x) + x $.
$endgroup$
– Ben
Dec 7 '18 at 16:23
add a comment |
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$begingroup$
I believe we still need the same assumptions Fubini requires. Is $f$ a functoom of a single variable?
$endgroup$
– Sean Roberson
Dec 7 '18 at 4:44
$begingroup$
Function $ f $ is a function of $ g(x) $ and $ x $. For example, the simplest one is $ f(x) = g(x) + x $.
$endgroup$
– Ben
Dec 7 '18 at 16:23