How to show that $C$ is countable?

Question

Let $C=\{B_a:a\in A\}$ be a collection of piecewise disjoint measurable subsets of $[0,1]$ having positive Lebesgue measure.

You may have a look at the end of this answer: http://math.stackexchange.com/questions/74676/axiom-of-choice-non-measurable-sets-countable-unions/74679#74679 — Olivier Oloa, May 10 '15 at 14:04

score 5 · Accepted Answer · answered May 10 '15 at 13:54

5

Let $C_n := \{B\in C~\vert~ \lambda(B) \geq \frac{1}{n}\}$. Then $C$ is the union of all $C_n$ and $C_n$ is finite for each $n$ (the cardinality is bounded by $n$ to be precise).

answered May 10 '15 at 13:54

Lukas Betz

4,506

why is $C_n$ finite? – Learnmore May 10 '15 at 13:56
what if there exist more than $n$ sets in $C_n$ ? Note that they are disjoint. – Lukas Betz May 10 '15 at 13:57
can you please elaborate? – Learnmore May 10 '15 at 13:58
$\displaystyle 1\geq \lambda(\bigcup C_n) = \sum_{B\in C_n} \lambda(B) \geq \sum _{B\in C_n}\frac{1}{n} = \frac{1}{n}|C_n|$ – Lukas Betz May 10 '15 at 14:01
thanks but why is $\lambda(\cup C_n)\leq 1$? – Learnmore May 11 '15 at 00:28
because the union is a subset of [0,1] – Lukas Betz May 11 '15 at 00:29
thank you very much for the answer – Learnmore May 11 '15 at 00:42

How to show that $C$ is countable?

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