Mixing
How to show, that a function defined on a product $sigma$-algebra is measurable.
$begingroup$
Consider measurable spaces $(A_1,mathcal{A}_1)$, $(A_2,mathcal{A}_2)$ and their product $sigma$-algebra $(A_1times A_2,mathcal{A}_1otimes mathcal{A}_2)$. I want to show, that
$$f:A_1times A_2longrightarrowmathbb{R}$$
is measurable. Is it sufficient, to show that for any $a_1in A_1$, $a_2in A_2$ the functions
$$f(cdot,a_2):A_1longrightarrowmathbb{R}$$
$$f(a_1,cdot):A_2longrightarrowmathbb{R}$$
are measurable?
real-analysis analysis measure-theory
edited Jan 20 at 3:41
J. W. Tanner
2,4831117
asked Jan 17 at 18:48
user408858user408858
482213
$endgroup$
add a comment |
$begingroup$
Consider measurable spaces $(A_1,mathcal{A}_1)$, $(A_2,mathcal{A}_2)$ and their product $sigma$-algebra $(A_1times A_2,mathcal{A}_1otimes mathcal{A}_2)$. I want to show, that
$$f:A_1times A_2longrightarrowmathbb{R}$$
is measurable. Is it sufficient, to show that for any $a_1in A_1$, $a_2in A_2$ the functions
$$f(cdot,a_2):A_1longrightarrowmathbb{R}$$
$$f(a_1,cdot):A_2longrightarrowmathbb{R}$$
are measurable?
real-analysis analysis measure-theory
edited Jan 20 at 3:41
J. W. Tanner
2,4831117
asked Jan 17 at 18:48
user408858user408858
482213
$endgroup$
1
$begingroup$
It is not sufficient, see David Mitra's comment to the question math.stackexchange.com/q/647235/27978 (the book referenced, Gelbaum and Olmsted, "Counterexamples in Analysis" is well worth having in your library if real analysis is your thing).
$endgroup$
– copper.hat
Jan 23 at 17:58
add a comment |
$begingroup$
Consider measurable spaces $(A_1,mathcal{A}_1)$, $(A_2,mathcal{A}_2)$ and their product $sigma$-algebra $(A_1times A_2,mathcal{A}_1otimes mathcal{A}_2)$. I want to show, that
$$f:A_1times A_2longrightarrowmathbb{R}$$
is measurable. Is it sufficient, to show that for any $a_1in A_1$, $a_2in A_2$ the functions
$$f(cdot,a_2):A_1longrightarrowmathbb{R}$$
$$f(a_1,cdot):A_2longrightarrowmathbb{R}$$
are measurable?
real-analysis analysis measure-theory
edited Jan 20 at 3:41
J. W. Tanner
2,4831117
asked Jan 17 at 18:48
user408858user408858
482213
$endgroup$
Consider measurable spaces $(A_1,mathcal{A}_1)$, $(A_2,mathcal{A}_2)$ and their product $sigma$-algebra $(A_1times A_2,mathcal{A}_1otimes mathcal{A}_2)$. I want to show, that
$$f:A_1times A_2longrightarrowmathbb{R}$$
is measurable. Is it sufficient, to show that for any $a_1in A_1$, $a_2in A_2$ the functions
$$f(cdot,a_2):A_1longrightarrowmathbb{R}$$
$$f(a_1,cdot):A_2longrightarrowmathbb{R}$$
are measurable?
real-analysis analysis measure-theory
real-analysis analysis measure-theory
edited Jan 20 at 3:41
J. W. Tanner
2,4831117
asked Jan 17 at 18:48
user408858user408858
482213
edited Jan 20 at 3:41
J. W. Tanner
2,4831117
asked Jan 17 at 18:48
user408858user408858
482213
edited Jan 20 at 3:41
J. W. Tanner
2,4831117
edited Jan 20 at 3:41
J. W. Tanner
2,4831117
edited Jan 20 at 3:41
J. W. Tanner
2,4831117
2,4831117
asked Jan 17 at 18:48
user408858user408858
482213
asked Jan 17 at 18:48
user408858user408858
482213
asked Jan 17 at 18:48
user408858user408858
482213
482213
1
$begingroup$
It is not sufficient, see David Mitra's comment to the question math.stackexchange.com/q/647235/27978 (the book referenced, Gelbaum and Olmsted, "Counterexamples in Analysis" is well worth having in your library if real analysis is your thing).
$endgroup$
– copper.hat
Jan 23 at 17:58
add a comment |
1
$begingroup$
It is not sufficient, see David Mitra's comment to the question math.stackexchange.com/q/647235/27978 (the book referenced, Gelbaum and Olmsted, "Counterexamples in Analysis" is well worth having in your library if real analysis is your thing).
$endgroup$
– copper.hat
Jan 23 at 17:58
1
1
$begingroup$
It is not sufficient, see David Mitra's comment to the question math.stackexchange.com/q/647235/27978 (the book referenced, Gelbaum and Olmsted, "Counterexamples in Analysis" is well worth having in your library if real analysis is your thing).
$endgroup$
– copper.hat
Jan 23 at 17:58
$begingroup$
It is not sufficient, see David Mitra's comment to the question math.stackexchange.com/q/647235/27978 (the book referenced, Gelbaum and Olmsted, "Counterexamples in Analysis" is well worth having in your library if real analysis is your thing).
$endgroup$
– copper.hat
Jan 23 at 17:58
add a comment |
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$begingroup$
It is not sufficient, see David Mitra's comment to the question math.stackexchange.com/q/647235/27978 (the book referenced, Gelbaum and Olmsted, "Counterexamples in Analysis" is well worth having in your library if real analysis is your thing).
$endgroup$
– copper.hat
Jan 23 at 17:58