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

How to show that $C$ is countable?

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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).

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  • $\begingroup$ why is $C_n$ finite? $\endgroup$ – Learnmore May 10 '15 at 13:56
  • $\begingroup$ what if there exist more than $n$ sets in $C_n$ ? Note that they are disjoint. $\endgroup$ – Tim B. May 10 '15 at 13:57
  • $\begingroup$ can you please elaborate? $\endgroup$ – Learnmore May 10 '15 at 13:58
  • $\begingroup$ $\displaystyle 1\geq \lambda(\bigcup C_n) = \sum_{B\in C_n} \lambda(B) \geq \sum _{B\in C_n}\frac{1}{n} = \frac{1}{n}|C_n|$ $\endgroup$ – Tim B. May 10 '15 at 14:01
  • $\begingroup$ thanks but why is $\lambda(\cup C_n)\leq 1$? $\endgroup$ – Learnmore May 11 '15 at 0:28

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