Lebesgue measure space is the real line associated with Lebesgue measure for all its Lebesgue measurable subsets, symbolically denoted by $( \mathbb{R}, \mathcal{L}, \mathbf{m} )$. It is the typical measure space on the real line.
Lebesgue measure is derived from interval length.
Lebesgue measure can be derived from completion of the Borel measure. It includes all subsets of Borel sets of measure zero, which are not countable unions and intersections of Borel sets.
A complete measure space is a measure space in which every subset of every null set is measurable. It follows that those sets have measure zero.
Symbolically, measure space $(X, \sigma, \mu)$ is complete iff $S \subseteq N \in \Sigma, \mu(N) = 0 \Rightarrow S \in \Sigma$
The completion of a measure space is the smallest extension of the measure space that is complete.
Symbolically, given a measure space $(X, \Sigma, \mu)$:
Then, measure space $(X, \Sigma_0, \mu_0)$ is the completion of $(X, \Sigma, \mu)$.
Fubini Theorem establishes a connection between multiple integral and iterated integrals.
Thm (Fubini): Given $(S,\mathcal{S},\mu)$ and $(T,\mathcal{T},\theta)$ are sigma-finite measure spaces, and $f: S\times T \to \mathbb{R}$ is a $(\mathcal{S} \times \mathcal{T}, \mathcal{B})$-measurable function. If $f\geq 0$ or $\int_S \, \mathrm{d} \mu \left(\int_T |f| \,\mathrm{d} \theta \right) < \infty$, then:
$$\int_{S \times T} f \,\mathrm{d}(\mu \times \theta) = \int_S \, \mathrm{d} \mu \left(\int_T f \,\mathrm{d} \theta \right) = \int_T \,\mathrm{d} \theta \left( \int_S f \, \mathrm{d} \mu \right)$$