Understanding Lebesgue Measure: m([a,b))=m((a,b])?

[pivoxa15]

Homework Statement

m((a,b])=b-a is defined as the lebesuge measure

what is m([a,b))?

The Attempt at a Solution

m({a})=0 for any a in R?

so m([a,b))=m((a,b])?

 

[Dick]

Sure...

 

[HallsofIvy]

If this is a course in measure theory, I'm sure the problems will get harder!

 

[pivoxa15]

They didn't specify that m({a})=0 for any a in R. So it was more a problem, of ambiguity.

It was part of a bigger problem.

 

[matt grime]

They did specify that m(pt)=0 - it is deducible from your first post. A point pt lies in any interval (pt -e/2 , pt+e/2] for any e, hence m(pt)<e for all e, thus it is zero.

 

[pivoxa15]

They did specify that m(pt)=0 - it is deducible from your first post. A point pt lies in any interval (pt -e/2 , pt+e/2] for any e, hence m(pt)<e for all e, thus it is zero.

Good point or maybe more easily it can be worked out from letting b=a

m((a,b])=b-a is defined as the lebesuge measure

=> m((a,a])=a-a=0
=> m(pt)=0

 

[matt grime]

(a.a] does not equal the set {a}. So what the measure of the (empty) set (a,a] is does not tell you what the measure of the non-empty set {a} is. (Even assuming that a one point set is measurable, of course.)

 

[pivoxa15]

(a,a] dosen't make sense does it. It should be lim n->infinity(a-1/n,a]={a}

 

[matt grime]

Taking limits of sets needs some careful consideration. Do you mean direct or inverse limit? It's the inverse limit, by the way.

If I were you I'd not attempt to write things like: the limit of these sets is that set. Stick to sequences of numbers, not sequences of sets.

 

[mikecon0523]

Measure of an interval

Homework Statement

m((a,b])=b-a is defined as the lebesuge measure

what is m([a,b))?

Ans: still b-a. ...

The Attempt at a Solution

m({a})=0 for any a in R?

so m([a,b))=m((a,b])?

1.
Ans: still b-a. ...