Lebesgue Measure Given a Subset and a Singleton Set in R
Question:
For a subset E of R and a number a ∈ R, let a+E = {a+e | e ∈ E}. Show that E is measurable if and only if a+E is measurable.
My approach:
==>
Suppose E is L. measurable.
Then a+E = a union E
SInce {a} in R and E both are L. mble, its union also is L. mble.
Hence, E is L.mble ==> a+E is L.mble -->(1)
Since (1) is true for any a in R and any subset E of R, replace E and a+E in (1) by a+E and E respectively to get,
a+E is L.mble ==> E is L.mble --->(2)
So from 1 and 2, E is L.mble iff a+E is L.mble
Hence the proof.
Is this a correct approach?
Thanks in advance for your help!
For a subset E of R and a number a ∈ R, let a+E = {a+e | e ∈ E}. Show that E is measurable if and only if a+E is measurable.
My approach:
==>
Suppose E is L. measurable.
Then a+E = a union E
SInce {a} in R and E both are L. mble, its union also is L. mble.
Hence, E is L.mble ==> a+E is L.mble -->(1)
Since (1) is true for any a in R and any subset E of R, replace E and a+E in (1) by a+E and E respectively to get,
a+E is L.mble ==> E is L.mble --->(2)
So from 1 and 2, E is L.mble iff a+E is L.mble
Hence the proof.
Is this a correct approach?
Thanks in advance for your help!
Original post by Ash760
Then a+E = a union E
Is this a correct approach?
Is this a correct approach?
You might like to check that statement; try some simple examples if you're unsure.
Original post by ghostwalker
You might like to check that statement; try some simple examples if you're unsure.
I've noticed my mistake.
This means that a is added to each element in E right?
Original post by Ash760
I've noticed my mistake.
This means that a is added to each element in E right?
This means that a is added to each element in E right?
Yep.
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