# Show that the complement of a non-measurable set is also n.m

1. Sep 25, 2016

### cragar

1. The problem statement, all variables and given/known data
Show that the complement of a non-measurable set is also non-measurable.
3. The attempt at a solution
Let A be a set that is non measurable. Let B be the complement of A.
Lets assume for contradiction that B has a measure. Now from the axioms of measure theory they have countable additive of measure. case 1: A+B has a measure.
If B has a measure and A+B has a measure then A+B-B should have a measure. But A has no measure so this is a contradiction therefore B has no measure.
Case 2: A+B has no measure.
Lets assume B has a measure, If B has a measure then A+B has a measure
But this is a contradiction, therefore B has no measure .
Not sure If on case 2 I can talk about A+B having a measure

2. Sep 25, 2016

### andrewkirk

It depends on the definition of measurable used. The usual definition is as shown here. Under that definition, all sets in the sigma-algebra $\Sigma$ are measurable and, since a set is in $\Sigma$ iff its complement is, the result follows trivially.

I suspect you are using a different definition of measurable - perhaps a specific measure like Lebesgue measure. There is a list of possible measures here. Do you mean one of them? If so, which one?

3. Sep 26, 2016

### cragar

When my teacher told us about the complement of a non-measurable set is non-measurable. I wanted to prove this for myself, he talked about it before Lebesgue measure. And before sigma-algebra. We just had the basic axioms of measure theory like countably additive, the measure of a point is zero. the measure of an interval is the difference between the endpoints. Like for example the measure of the irrationals is infinite.

4. Sep 26, 2016

### micromass

Staff Emeritus
A measure is a function. What is the domain of the measure defined like?

5. Sep 26, 2016

### cragar

It can be no more than a countable union of disjoint sets. If the measure exists it will be assigned a non-negative real number. And it is closed under compliment.
If its closed under compliment then the set and its compliment will have a measure. Because the set and its compliment is just one partition of the set. thanks for the responses