Show there is no measure on $\mathbb{N}$ such that $\mu(\{0,k,2k,\ldots\})=\frac{1}{k}$ for all $k\ge 1$

For $k\ge1$, let $A_k=\{0,k,2k,\ldots\}.$ Show that there is no measure $\mu$ on $\mathbb{N}$ satisfying $\mu(A_k)=\frac{1}{k}$ for all $k\ge1$.

What I have done so far:
I am trying to apply Borel-Cantelli lemma ($\mu(\mathbb{N})=1)$. Let $(p_n)_{n\in \mathbb{N}}=(2,3,5,\ldots)$ be the increasing sequence of all prime numbers. It is the case that for all $k\in\mathbb{N}$ and $i_1\lt i_2 \lt \ldots \lt i_k$ we have $\mu\left(A_{p_{i_1}}\cap\ldots\cap A_{p_{i_k}}\right)=\mu\left(A_{p_{i_1}}\right)\ldots\mu\left(A_{p_{i_k}}\right)$ so all $A_n$ are independent. We know that the sum of the reciprocals of the primes $\sum\limits_{n=1}^\infty \frac{1}{p_n}$diverges and hence, by B-C second lemma, $$\mu\left(\bigcap\limits_{n=1}^\infty\bigcup\limits_{m=n}^\infty A_{p_m}\right)=1$$ holds. However, I cannot figure out how to conclude the reasoning. I would appreciate any help.

-

1 Answer

You are basically there. You proved in your last line that $\mu$-a.e. number is divisible by infinitely many primes, which is an obvious contradiction.

-

Oh, ok. I am almost sure I understand. Is the following $\mathbb{N}\subset \bigcup\limits_{n=1}^\infty\bigcap\limits_{m=n}^\infty A_{p_m}^c$ the precise form of what you have just stated? Also it is an easy contradiction, because the whole space cannot be contained in the set of measure $0$. Am I correct? – Kuba Helsztyński Oct 29 '12 at 0:52
Yes, this is correct. – Lukas Geyer Oct 29 '12 at 1:55
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 19, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9589319229125977, "perplexity_flag": "head"}