How to do Lebesgue decomposition in this problem?

$\{x_n\}$ is a sequence in $\mathbb{R}$ and $\{p_n\}$ is a sequence of positive numbers. Define a $\sigma$-finite measure $\nu(E)=\sum_{x_n\in E}p_n$. Find the Lebesgue decomposition of $\nu$ with respect to Lebesgue measure on $\mathbb{R}$.

My solution:

Let $S=\{x_n\}_{n\in\mathbb{N}}$. Assume $\lambda(E)=0,\forall E\subseteq \mathbb{R}$.

Define $\nu_1(E)=\nu(E\backslash S)=0$ because $E\backslash S$ does not have corresponding $\{p_n\}\Rightarrow \nu_1\ll\lambda$.

Define $\nu_2(E)=\nu(E\cap S)=\sum_{x_n\in E}p_n>0\Rightarrow \nu_2\perp\lambda$.

Any comment is appreciated.

-

1
You cannot assume $\lambda(E) = 0 \quad \forall\, E \subseteq \mathbb{R}$. I suppose that is not what you wanted to write. – Thomas Dec 10 '12 at 9:15
Thank you, @Thomas. – Sam Dec 10 '12 at 15:20
{"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": 14, "mathjax_display_tex": 0, "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": false, "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": 0.9066641926765442, "perplexity_flag": "head"}