Mixing
A question about proving that $f(x,y)$ is Integrable on $A subseteq mathbb R^2$
$begingroup$
so i'm looking at the following theorem which i want to prove:
$f(x,y)$ continuous and bounded on a bounded measurable set $A subseteq R^2$ $Rightarrow$ $f(x,y)$ is integrable on $A$
So my question is not that i don't know where to start. It's pretty much the same as in $mathbb R$ but am i not missing the assumption that either $f(x,y)$ is bounded on $overline A$ (the closure of $A$), or that $A$ is a closed set?
integration multivariable-calculus proof-verification proof-writing proof-explanation
asked Jan 6 at 19:00
strangeattractorstrangeattractor
497
$endgroup$
add a comment |
$begingroup$
so i'm looking at the following theorem which i want to prove:
$f(x,y)$ continuous and bounded on a bounded measurable set $A subseteq R^2$ $Rightarrow$ $f(x,y)$ is integrable on $A$
So my question is not that i don't know where to start. It's pretty much the same as in $mathbb R$ but am i not missing the assumption that either $f(x,y)$ is bounded on $overline A$ (the closure of $A$), or that $A$ is a closed set?
integration multivariable-calculus proof-verification proof-writing proof-explanation
asked Jan 6 at 19:00
strangeattractorstrangeattractor
497
$endgroup$
$begingroup$
Why do you care about $bar A$ or $A$ being closed?
$endgroup$
– zhw.
Jan 6 at 19:23
$begingroup$
I'm not sure about the boundary of A...isn't it possible for f to diverge to infintity as it approaches the boundary of A and mess up the integral? or does the fact that it is bounded on A prohibit such thing from occurring?
$endgroup$
– strangeattractor
Jan 6 at 20:24
add a comment |
$begingroup$
so i'm looking at the following theorem which i want to prove:
$f(x,y)$ continuous and bounded on a bounded measurable set $A subseteq R^2$ $Rightarrow$ $f(x,y)$ is integrable on $A$
So my question is not that i don't know where to start. It's pretty much the same as in $mathbb R$ but am i not missing the assumption that either $f(x,y)$ is bounded on $overline A$ (the closure of $A$), or that $A$ is a closed set?
integration multivariable-calculus proof-verification proof-writing proof-explanation
asked Jan 6 at 19:00
strangeattractorstrangeattractor
497
$endgroup$
so i'm looking at the following theorem which i want to prove:
$f(x,y)$ continuous and bounded on a bounded measurable set $A subseteq R^2$ $Rightarrow$ $f(x,y)$ is integrable on $A$
So my question is not that i don't know where to start. It's pretty much the same as in $mathbb R$ but am i not missing the assumption that either $f(x,y)$ is bounded on $overline A$ (the closure of $A$), or that $A$ is a closed set?
integration multivariable-calculus proof-verification proof-writing proof-explanation
integration multivariable-calculus proof-verification proof-writing proof-explanation
asked Jan 6 at 19:00
strangeattractorstrangeattractor
497
asked Jan 6 at 19:00
strangeattractorstrangeattractor
497
asked Jan 6 at 19:00
strangeattractorstrangeattractor
497
asked Jan 6 at 19:00
strangeattractorstrangeattractor
497
asked Jan 6 at 19:00
strangeattractorstrangeattractor
497
497
$begingroup$
Why do you care about $bar A$ or $A$ being closed?
$endgroup$
– zhw.
Jan 6 at 19:23
$begingroup$
I'm not sure about the boundary of A...isn't it possible for f to diverge to infintity as it approaches the boundary of A and mess up the integral? or does the fact that it is bounded on A prohibit such thing from occurring?
$endgroup$
– strangeattractor
Jan 6 at 20:24
add a comment |
$begingroup$
Why do you care about $bar A$ or $A$ being closed?
$endgroup$
– zhw.
Jan 6 at 19:23
$begingroup$
I'm not sure about the boundary of A...isn't it possible for f to diverge to infintity as it approaches the boundary of A and mess up the integral? or does the fact that it is bounded on A prohibit such thing from occurring?
$endgroup$
– strangeattractor
Jan 6 at 20:24
$begingroup$
Why do you care about $bar A$ or $A$ being closed?
$endgroup$
– zhw.
Jan 6 at 19:23
$begingroup$
Why do you care about $bar A$ or $A$ being closed?
$endgroup$
– zhw.
Jan 6 at 19:23
$begingroup$
I'm not sure about the boundary of A...isn't it possible for f to diverge to infintity as it approaches the boundary of A and mess up the integral? or does the fact that it is bounded on A prohibit such thing from occurring?
$endgroup$
– strangeattractor
Jan 6 at 20:24
$begingroup$
I'm not sure about the boundary of A...isn't it possible for f to diverge to infintity as it approaches the boundary of A and mess up the integral? or does the fact that it is bounded on A prohibit such thing from occurring?
$endgroup$
– strangeattractor
Jan 6 at 20:24
add a comment |
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$begingroup$
Why do you care about $bar A$ or $A$ being closed?
$endgroup$
– zhw.
Jan 6 at 19:23
$begingroup$
I'm not sure about the boundary of A...isn't it possible for f to diverge to infintity as it approaches the boundary of A and mess up the integral? or does the fact that it is bounded on A prohibit such thing from occurring?
$endgroup$
– strangeattractor
Jan 6 at 20:24