Exercise from David Williams' "Probability with Martingales" page 16

Asked Jul 06 '16 at 22:44

Active Jul 06 '16 at 23:12

Viewed 353 times

I am reading David Williams' "Probability with Martingales" and I'm completely stuck at the exercise page 16:

Let $\mathcal{C}$ be the class of subsets $C$ of $\mathbb{N}$ for which $$\lim_{m \uparrow \infty}m^{-1}\#\{k:1 \leq k \leq m; k \in C\}$$ exists. You should find elements $F$ and $G$ in $\mathcal{C}$ for which $F \cap G \notin \mathcal{C}$.

(I am able to find one subset of $\mathbb{N}$ that doesn't belong to $\mathcal{C}$, but unable to find anything like $F$ and $G$)

edited Jul 06 '16 at 23:12

asked Jul 06 '16 at 22:44

Frédéric Delanchy

Should it be $m$ goes to infinity? – jdods Jul 06 '16 at 22:47
What if you "expand" your set $A$ that does not belong to $C$ by considering $2A$ (so all its elements are even), then consider $F$ and $G$ to each have limiting measure $1/2$ and $F\cap G = 2A$? – Michael Jul 06 '16 at 23:45
How do you define $F$ and $G$ to end up with a limiting measure $1/2$ ? – Frédéric Delanchy Jul 07 '16 at 22:06
A standard construction for such a set is available in the accepted answer here – stochasticboy321 Jul 07 '16 at 23:55
Thank you, this is the exact answer to my question – Frédéric Delanchy Jul 08 '16 at 20:33

Exercise from David Williams' "Probability with Martingales" page 16

0 Answers0