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 $\mathcal{B}(\mathbb{R})$?