MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document We have the...

Question

Sattelhofsk

Answered question

2022-06-21

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document We have the outer measure

μ^{*} (A) = inf {\sum_{n = 1}^{\infty} μ_{0} (A_{n}) | A_{n} \in A_{0}, A \subset ⋃_{n = 1}^{\infty} A_{n}}

The inner measure is:

μ_{*} (A) = μ_{0} (X) - μ^{*} (X ∖ A)

Where

μ_{0} (A) = \sum_{n = 1}^{\infty} μ_{0} (A_{n})

and

A_{0}

is a

σ

algebra.
But why is

μ_{*} \leq μ^{*}

?
I always end up with a contradiction.
My proof would be:

μ_{*} (A) - μ^{*} (A) = μ_{0} (X) - μ^{*} (X ∖ A) - μ^{*} (A)

Because:

μ^{*} (A) = μ_{0} (A) - μ_{*} (\emptyset)

We get:

μ_{*} (A) - μ^{*} (A) = μ_{0} (X) - μ^{*} (X ∖ A) - μ_{0} (A) + μ_{*} (\emptyset)

μ_{*} (A) - μ^{*} (A) = μ_{0} (X ∖ A) - μ^{*} (X ∖ A)

Because by definition

μ^{*} (A) = inf \sum_{n = 1}^{\infty} μ_{0} (A_{n}) \leq \sum_{n = 1}^{\infty} μ_{0} (A_{n})

this means, that

μ_{0} (X ∖ A) - μ^{*} (X ∖ A) \geq 0

We get:

μ_{*} (A) - μ^{*} (A) \geq 0

μ_{*} (A) \geq μ^{*} (A)

So where's my mistake?