What is Lebesgue measure on ℝᵈ?

Lebesgue measure on $\R^d$ is $\lambda_d = \lambda\otimes\cdots\otimes\lambda$ ($d$ factors; associativity of the construction is checked on boxes and propagated by uniqueness). It is the unique Borel measure giving each box $\prod\intoc{a_i}{b_i}$ its volume $\prod(b_i - a_i)$; it is translation-invariant (translates agree on boxes), $\sigma$-finite, and complete after Carathéodory completion — we write $\lambda_d$ for the completed measure and integrate accordingly.