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The Compromise

What Vitali’s impossibility proof shows is that there is no function m which has all of the following properties.

1. Nonnegative

2. Translation invariant

3. Assigns intervals their length

4. Countably additive

5. Defined for all subsets of \(\Bbb R\)

We may therefore “block” the proof by relinquishing one of these. Numbers 1 through 3 are simply non-negotiable. I believe some mathematicians have made hay of giving up number (4).

However the main decision that we study in measure theory is giving up number 5. Therefore, we will eventually distinguish a sub-collection of \(\mathcal P(\Bbb R)\) which we will regard as the measurable subsets.