notes

## Lebesgue Measure

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.

### Complete measure space

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 $S \subseteq N \in \Sigma, \mu(N) = 0 \Rightarrow S \in \Sigma$

### Completion of measure space

The completion of a measure space is the smallest extension of the measure space that is complete.

Symbolically, given a measure space $(X, \Sigma, \mu)$:

1. Construct the set of all subsets of $\mu$-measure zero subsets of X, denote as $Z$.
2. Generate the sigma-algebra from $\Sigma$ and $Z$, denote as $\Sigma_0$.
3. Extend the measure $\mu$ to $\Sigma_0$, such that $\forall C \in \Sigma_0, \mu_0 (C) = \inf \{ \mu (D) | C \subseteq D \in \Sigma \}$

Then, measure space $(X, \Sigma_0, \mu_0)$ is the completion of $(X, \Sigma, \mu)$.

## Integration

### Integration on product space

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:

$$\int_{S \times T} f \,\mathrm{d}(\mu \times \theta) = \int_S \, \mathrm{d} \mu \left(\int_T f \,\mathrm{d} \theta \right) = \int_T \,\mathrm{d} \theta \left( \int_S f \, \mathrm{d} \mu \right)$$