# Problem with the definition of Lebesgue Measure and borel sets!

The definition of lebesgue measure (in my textbook):

The set-function $\lambda^{n}$ on ($\mathbb{R}^{n}, \mathcal{B}(\mathbb{R}^{n})$) that assigns every half-open $[[a,b)) = [a_{1},b_{1}) \times \dots \times [a_{n},b_{n})\in\mathcal{J}$ the value: $\lambda^{n}([[a,b))):=\prod_{j=1}^{n}(b_{j}-a_{j})$ is called n-dimensional Lebesgue measure.

On the next page the book mentions that the Lebesque Measure is a measure on the Borel sets $\mathcal{B}(\mathbb{R})$

My question/problem:

According to the definition of a measure in the book. The measure must be a map between $\mathcal{B}(\mathbb{R})$ and $[0,\infty]$. But how can the defined $\lambda^{n}$ maps all the elements in $\mathcal{B}(\mathbb{R})$? When it only maps half-open rectangles? Can $\lambda^{n}$ map other kinds of set in $\mathcal{B}(\mathbb{R})$?

-

4
This is not only a definition, but also a theorem : there exists a unique measure on Borel sets which gives this value to each half-open rectangles. This is not really obvious, and is a consequence of Caratheodory's extension theorem. You should find details in textobooks on measure theory. – Ahriman Sep 9 '12 at 14:43
Usually, the Lebesgue measure is defined not only on Borel sets but also on all Lebesgue-measurabe sets. Note that the measure must be $\sigma$-additive. If you define it only on sets $[[a,b))$, its value on other sets will be defined uniquely. – Yury Sep 9 '12 at 14:52
– t.b. Sep 10 '12 at 10:10
{"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": 13, "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": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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": 0.9186446070671082, "perplexity_flag": "head"}