Showing that a multivariable function is Borel measurable. [closed]
$begingroup$
I want to show that the following function is Borel measurable. Consider:
$$f: mathbb{R}^2 rightarrow mathbb{R}:left{
begin{array}{ll}
sinleft(frac{1}{x-y}right) & x> y \
x^2 + y^2 & xleq y. \
end{array}
right. $$
So I think one can prove it by looking if the function is continuous (so measurable) or proving it by the defintition of a Borel measurable function. By proving it from the defenition, one can use that ${x:f(x)>a}$ is a Borel set. But I don't know how to proceed with this defenition.
functional-analysis measure-theory borel-sets
asked Jan 5 at 19:05
Belgium_PhysicsBelgium_Physics
325110
$endgroup$
closed as off-topic by Nosrati, Paul Frost, Cesareo, José Carlos Santos, amWhy Jan 6 at 20:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Paul Frost, Cesareo, José Carlos Santos, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I want to show that the following function is Borel measurable. Consider:
$$f: mathbb{R}^2 rightarrow mathbb{R}:left{
begin{array}{ll}
sinleft(frac{1}{x-y}right) & x> y \
x^2 + y^2 & xleq y. \
end{array}
right. $$
So I think one can prove it by looking if the function is continuous (so measurable) or proving it by the defintition of a Borel measurable function. By proving it from the defenition, one can use that ${x:f(x)>a}$ is a Borel set. But I don't know how to proceed with this defenition.
functional-analysis measure-theory borel-sets
asked Jan 5 at 19:05
Belgium_PhysicsBelgium_Physics
325110
$endgroup$
closed as off-topic by Nosrati, Paul Frost, Cesareo, José Carlos Santos, amWhy Jan 6 at 20:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Paul Frost, Cesareo, José Carlos Santos, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I want to show that the following function is Borel measurable. Consider:
$$f: mathbb{R}^2 rightarrow mathbb{R}:left{
begin{array}{ll}
sinleft(frac{1}{x-y}right) & x> y \
x^2 + y^2 & xleq y. \
end{array}
right. $$
So I think one can prove it by looking if the function is continuous (so measurable) or proving it by the defintition of a Borel measurable function. By proving it from the defenition, one can use that ${x:f(x)>a}$ is a Borel set. But I don't know how to proceed with this defenition.
functional-analysis measure-theory borel-sets
asked Jan 5 at 19:05
Belgium_PhysicsBelgium_Physics
325110
$endgroup$
I want to show that the following function is Borel measurable. Consider:
$$f: mathbb{R}^2 rightarrow mathbb{R}:left{
begin{array}{ll}
sinleft(frac{1}{x-y}right) & x> y \
x^2 + y^2 & xleq y. \
end{array}
right. $$
So I think one can prove it by looking if the function is continuous (so measurable) or proving it by the defintition of a Borel measurable function. By proving it from the defenition, one can use that ${x:f(x)>a}$ is a Borel set. But I don't know how to proceed with this defenition.
functional-analysis measure-theory borel-sets
functional-analysis measure-theory borel-sets
asked Jan 5 at 19:05
Belgium_PhysicsBelgium_Physics
325110
asked Jan 5 at 19:05
Belgium_PhysicsBelgium_Physics
325110
asked Jan 5 at 19:05
Belgium_PhysicsBelgium_Physics
325110
asked Jan 5 at 19:05
Belgium_PhysicsBelgium_Physics
325110
asked Jan 5 at 19:05
Belgium_PhysicsBelgium_Physics
325110
325110
closed as off-topic by Nosrati, Paul Frost, Cesareo, José Carlos Santos, amWhy Jan 6 at 20:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Paul Frost, Cesareo, José Carlos Santos, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Nosrati, Paul Frost, Cesareo, José Carlos Santos, amWhy Jan 6 at 20:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Paul Frost, Cesareo, José Carlos Santos, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
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