How to find two functions are measuarable or not?
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Let $E$ be a non measurable subset of $(0,1)$. Define two functions as follows:
$f_1(x)=1/x$ if $x∈E$ and $0$ if $x∉E$ and
$f_2(x)=0$ if $x∈E $ and $f(x)=1/x$ if $x∉E$
then are $f_1$ and $f_2$ measurable ?
measure-theory
asked Sep 22 '17 at 9:27
rohi
214
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up vote
0
down vote
favorite
Let $E$ be a non measurable subset of $(0,1)$. Define two functions as follows:
$f_1(x)=1/x$ if $x∈E$ and $0$ if $x∉E$ and
$f_2(x)=0$ if $x∈E $ and $f(x)=1/x$ if $x∉E$
then are $f_1$ and $f_2$ measurable ?
measure-theory
asked Sep 22 '17 at 9:27
rohi
214
Please learn to use MathJax here. It makes it much easier to read your problem.
– Landuros
Sep 22 '17 at 9:30
I have changed the awful mess you had given into a readable text. Next time, to begin with, enclose mathematical formulas between dollar signs and learn MathJax.
– Jean Marie
Sep 22 '17 at 9:34
You should give your own attempt on the problem, before you can expect us to help you. To start: what is the definition of measurable function? What happens when you try to apply it to $f_1$?
– GEdgar
Sep 22 '17 at 10:45
By definition, an extended real valued function f is said to be Lebesgue measurable if its domain is measurable and for each real α the set {x: f(x) < α } is measurable.
– rohi
Sep 22 '17 at 11:03
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $E$ be a non measurable subset of $(0,1)$. Define two functions as follows:
$f_1(x)=1/x$ if $x∈E$ and $0$ if $x∉E$ and
$f_2(x)=0$ if $x∈E $ and $f(x)=1/x$ if $x∉E$
then are $f_1$ and $f_2$ measurable ?
measure-theory
asked Sep 22 '17 at 9:27
rohi
214
Let $E$ be a non measurable subset of $(0,1)$. Define two functions as follows:
$f_1(x)=1/x$ if $x∈E$ and $0$ if $x∉E$ and
$f_2(x)=0$ if $x∈E $ and $f(x)=1/x$ if $x∉E$
then are $f_1$ and $f_2$ measurable ?
measure-theory
measure-theory
asked Sep 22 '17 at 9:27
rohi
214
asked Sep 22 '17 at 9:27
rohi
214
edited Sep 22 '17 at 11:10
asked Sep 22 '17 at 9:27
rohi
214
asked Sep 22 '17 at 9:27
rohi
214
asked Sep 22 '17 at 9:27
rohi
214
214
Please learn to use MathJax here. It makes it much easier to read your problem.
– Landuros
Sep 22 '17 at 9:30
I have changed the awful mess you had given into a readable text. Next time, to begin with, enclose mathematical formulas between dollar signs and learn MathJax.
– Jean Marie
Sep 22 '17 at 9:34
You should give your own attempt on the problem, before you can expect us to help you. To start: what is the definition of measurable function? What happens when you try to apply it to $f_1$?
– GEdgar
Sep 22 '17 at 10:45
By definition, an extended real valued function f is said to be Lebesgue measurable if its domain is measurable and for each real α the set {x: f(x) < α } is measurable.
– rohi
Sep 22 '17 at 11:03
add a comment |
Please learn to use MathJax here. It makes it much easier to read your problem.
– Landuros
Sep 22 '17 at 9:30
I have changed the awful mess you had given into a readable text. Next time, to begin with, enclose mathematical formulas between dollar signs and learn MathJax.
– Jean Marie
Sep 22 '17 at 9:34
You should give your own attempt on the problem, before you can expect us to help you. To start: what is the definition of measurable function? What happens when you try to apply it to $f_1$?
– GEdgar
Sep 22 '17 at 10:45
By definition, an extended real valued function f is said to be Lebesgue measurable if its domain is measurable and for each real α the set {x: f(x) < α } is measurable.
– rohi
Sep 22 '17 at 11:03
Please learn to use MathJax here. It makes it much easier to read your problem.
– Landuros
Sep 22 '17 at 9:30
Please learn to use MathJax here. It makes it much easier to read your problem.
– Landuros
Sep 22 '17 at 9:30
I have changed the awful mess you had given into a readable text. Next time, to begin with, enclose mathematical formulas between dollar signs and learn MathJax.
– Jean Marie
Sep 22 '17 at 9:34
I have changed the awful mess you had given into a readable text. Next time, to begin with, enclose mathematical formulas between dollar signs and learn MathJax.
– Jean Marie
Sep 22 '17 at 9:34
You should give your own attempt on the problem, before you can expect us to help you. To start: what is the definition of measurable function? What happens when you try to apply it to $f_1$?
– GEdgar
Sep 22 '17 at 10:45
You should give your own attempt on the problem, before you can expect us to help you. To start: what is the definition of measurable function? What happens when you try to apply it to $f_1$?
– GEdgar
Sep 22 '17 at 10:45
By definition, an extended real valued function f is said to be Lebesgue measurable if its domain is measurable and for each real α the set {x: f(x) < α } is measurable.
– rohi
Sep 22 '17 at 11:03
By definition, an extended real valued function f is said to be Lebesgue measurable if its domain is measurable and for each real α the set {x: f(x) < α } is measurable.
– rohi
Sep 22 '17 at 11:03
add a comment |
1 Answer
1
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0
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Hint
What about the sets ${xin(0,1)colon f_1(x)>0}$ and ${xin(0,1)colon f_2(x)=0}$? Read about equivalent conditions of measurability.
Another, more theoretical approach: $f_1(x)=frac{1}{x}chi_E(x)$. The set is measurable iff its characteristic function is measurable. Similar argument could be applied for $f_2$.
answered Sep 22 '17 at 11:28
szw1710
6,3801123
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Hint
What about the sets ${xin(0,1)colon f_1(x)>0}$ and ${xin(0,1)colon f_2(x)=0}$? Read about equivalent conditions of measurability.
Another, more theoretical approach: $f_1(x)=frac{1}{x}chi_E(x)$. The set is measurable iff its characteristic function is measurable. Similar argument could be applied for $f_2$.
answered Sep 22 '17 at 11:28
szw1710
6,3801123
add a comment |
up vote
0
down vote
Hint
What about the sets ${xin(0,1)colon f_1(x)>0}$ and ${xin(0,1)colon f_2(x)=0}$? Read about equivalent conditions of measurability.
Another, more theoretical approach: $f_1(x)=frac{1}{x}chi_E(x)$. The set is measurable iff its characteristic function is measurable. Similar argument could be applied for $f_2$.
answered Sep 22 '17 at 11:28
szw1710
6,3801123
add a comment |
up vote
0
down vote
up vote
0
down vote
Hint
What about the sets ${xin(0,1)colon f_1(x)>0}$ and ${xin(0,1)colon f_2(x)=0}$? Read about equivalent conditions of measurability.
Another, more theoretical approach: $f_1(x)=frac{1}{x}chi_E(x)$. The set is measurable iff its characteristic function is measurable. Similar argument could be applied for $f_2$.
answered Sep 22 '17 at 11:28
szw1710
6,3801123
Hint
What about the sets ${xin(0,1)colon f_1(x)>0}$ and ${xin(0,1)colon f_2(x)=0}$? Read about equivalent conditions of measurability.
Another, more theoretical approach: $f_1(x)=frac{1}{x}chi_E(x)$. The set is measurable iff its characteristic function is measurable. Similar argument could be applied for $f_2$.
answered Sep 22 '17 at 11:28
szw1710
6,3801123
answered Sep 22 '17 at 11:28
szw1710
6,3801123
answered Sep 22 '17 at 11:28
szw1710
6,3801123
answered Sep 22 '17 at 11:28
szw1710
6,3801123
6,3801123
add a comment |
add a comment |
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Please learn to use MathJax here. It makes it much easier to read your problem.
– Landuros
Sep 22 '17 at 9:30
I have changed the awful mess you had given into a readable text. Next time, to begin with, enclose mathematical formulas between dollar signs and learn MathJax.
– Jean Marie
Sep 22 '17 at 9:34
You should give your own attempt on the problem, before you can expect us to help you. To start: what is the definition of measurable function? What happens when you try to apply it to $f_1$?
– GEdgar
Sep 22 '17 at 10:45
By definition, an extended real valued function f is said to be Lebesgue measurable if its domain is measurable and for each real α the set {x: f(x) < α } is measurable.
– rohi
Sep 22 '17 at 11:03