Lebesgue Measure equal to any translation invariant Measure on the Borel sigma Algebra
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I have come across a question in my measure theory textbook and can pretty much figure out how to construct an answer to the second part but not the first part. Any solutions or hints would be welcomed. The question is as follows:
Suppose $mu$ is a measure on $(mathbb{R}, B(mathbb{R}))$ such that $mu((0,1])=1$ and $mu$ is translation invariant. Here let $lambda$ denote the Lebesgue measure.
(i) Show that for every interval $A = (a,b]$ with $b-ainmathbb{Q}$, we have $mu(A)=lambda(A)$.
(ii) show that $mu=lambda$
Like I said I think I can do the second part by showing the the sets in part one which the measure agree on are a $pi$ system and then using the fact the Borel sets are generated by those intervals we can deduce the measure agree on the whole space, but I do not understand how to show they agree on the sets A in part (i).
Thanks in advance!
measure-theory lebesgue-measure borel-sets borel-measures
asked Nov 18 at 12:56
Dani_5040
373
add a comment |
up vote
0
down vote
favorite
I have come across a question in my measure theory textbook and can pretty much figure out how to construct an answer to the second part but not the first part. Any solutions or hints would be welcomed. The question is as follows:
Suppose $mu$ is a measure on $(mathbb{R}, B(mathbb{R}))$ such that $mu((0,1])=1$ and $mu$ is translation invariant. Here let $lambda$ denote the Lebesgue measure.
(i) Show that for every interval $A = (a,b]$ with $b-ainmathbb{Q}$, we have $mu(A)=lambda(A)$.
(ii) show that $mu=lambda$
Like I said I think I can do the second part by showing the the sets in part one which the measure agree on are a $pi$ system and then using the fact the Borel sets are generated by those intervals we can deduce the measure agree on the whole space, but I do not understand how to show they agree on the sets A in part (i).
Thanks in advance!
measure-theory lebesgue-measure borel-sets borel-measures
asked Nov 18 at 12:56
Dani_5040
373
See this.
– Math enthusiast
Nov 18 at 13:02
I don’t see how this relates as this Questions only deals with the Lebesgue Stieltjes Measure whereas my question is for a general one
– Dani_5040
Nov 19 at 10:38
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have come across a question in my measure theory textbook and can pretty much figure out how to construct an answer to the second part but not the first part. Any solutions or hints would be welcomed. The question is as follows:
Suppose $mu$ is a measure on $(mathbb{R}, B(mathbb{R}))$ such that $mu((0,1])=1$ and $mu$ is translation invariant. Here let $lambda$ denote the Lebesgue measure.
(i) Show that for every interval $A = (a,b]$ with $b-ainmathbb{Q}$, we have $mu(A)=lambda(A)$.
(ii) show that $mu=lambda$
Like I said I think I can do the second part by showing the the sets in part one which the measure agree on are a $pi$ system and then using the fact the Borel sets are generated by those intervals we can deduce the measure agree on the whole space, but I do not understand how to show they agree on the sets A in part (i).
Thanks in advance!
measure-theory lebesgue-measure borel-sets borel-measures
asked Nov 18 at 12:56
Dani_5040
373
I have come across a question in my measure theory textbook and can pretty much figure out how to construct an answer to the second part but not the first part. Any solutions or hints would be welcomed. The question is as follows:
Suppose $mu$ is a measure on $(mathbb{R}, B(mathbb{R}))$ such that $mu((0,1])=1$ and $mu$ is translation invariant. Here let $lambda$ denote the Lebesgue measure.
(i) Show that for every interval $A = (a,b]$ with $b-ainmathbb{Q}$, we have $mu(A)=lambda(A)$.
(ii) show that $mu=lambda$
Like I said I think I can do the second part by showing the the sets in part one which the measure agree on are a $pi$ system and then using the fact the Borel sets are generated by those intervals we can deduce the measure agree on the whole space, but I do not understand how to show they agree on the sets A in part (i).
Thanks in advance!
measure-theory lebesgue-measure borel-sets borel-measures
measure-theory lebesgue-measure borel-sets borel-measures
asked Nov 18 at 12:56
Dani_5040
373
asked Nov 18 at 12:56
Dani_5040
373
asked Nov 18 at 12:56
Dani_5040
373
asked Nov 18 at 12:56
Dani_5040
373
asked Nov 18 at 12:56
Dani_5040
373
373
See this.
– Math enthusiast
Nov 18 at 13:02
I don’t see how this relates as this Questions only deals with the Lebesgue Stieltjes Measure whereas my question is for a general one
– Dani_5040
Nov 19 at 10:38
add a comment |
See this.
– Math enthusiast
Nov 18 at 13:02
I don’t see how this relates as this Questions only deals with the Lebesgue Stieltjes Measure whereas my question is for a general one
– Dani_5040
Nov 19 at 10:38
See this.
– Math enthusiast
Nov 18 at 13:02
See this.
– Math enthusiast
Nov 18 at 13:02
I don’t see how this relates as this Questions only deals with the Lebesgue Stieltjes Measure whereas my question is for a general one
– Dani_5040
Nov 19 at 10:38
I don’t see how this relates as this Questions only deals with the Lebesgue Stieltjes Measure whereas my question is for a general one
– Dani_5040
Nov 19 at 10:38
add a comment |
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See this.
– Math enthusiast
Nov 18 at 13:02
I don’t see how this relates as this Questions only deals with the Lebesgue Stieltjes Measure whereas my question is for a general one
– Dani_5040
Nov 19 at 10:38