A key component of constructing the Lebesgue Integral is the Lebesgue measure which is a type of measure on the real numbers.
#### Generated \(\sigma\)-Algebra
Consider a set \(\Omega\) and a collection \(\mathcal{A}\) of subsets of \(\Omega\). We denote all [[sigma-algebra|\(\sigma\)-algebra]] containing \(\mathcal{A}\) by \(\mathcal{F}_{i},~i \in I\) where \(I\) is some index set. There exists a smallest \(\sigma\)-algebra containing \(\mathcal{A}\) denoted \(\sigma(\mathcal{A})\) - known as the [[generated-sigma-algebra|\(\sigma\)-algebra generated by \(\mathcal{A}\)]] - which is the intersection of all other \(\sigma\)-algebra containing \(\mathcal{A}\), i.e. \[
\sigma(\mathcal{A}) := \bigcap_{i\in I}\mathcal{F}_{i}.
\] \begin{proof} To show that \(\sigma(\mathcal{A})\) is a \(\sigma\)-algebra we simply note each of the following: 1. \(\forall i,\Omega \in\mathcal{F}_{i} \implies\Omega \in\cap_{i}\mathcal{F}_{i}\). 2. Let \(A\in\cap_{i}\mathcal{F}_{i}\implies \forall i, A\in\mathcal{F}_{i}\implies \forall i,A^c\in\mathcal{F}_{i}\implies A^c\in\cap_{i}\mathcal{F}_{i}\). 3. Let \(\{ A_{n} \}_{n\in\mathbb{N}}\subseteq \cap_{i}\mathcal{F}_{i}\implies \{ A_{n} \}_{n\in \mathbb{N}}\in \mathcal{F}_{i},~\forall i\in I\implies \bigcup_{n\in \mathbb{N}}A_{n}\in \mathcal{F}_{i},~ \forall i\in I\implies \bigcup_{n\in \mathbb{N}}A_{n}\in \bigcap_{i \in I}\mathcal{F}_{i}\). Hence \(\sigma(\mathcal{A})\) is a \(\sigma\)-algebra and since it is the intersection of all possible \(\sigma\)-algebra containing \(\mathcal{A}\) it is the smallest.\end{proof}
0.0.1 Measures on the Real Numbers
To construct measures on the real numbers \(\mathbb{R}\) we have to construct a \(\sigma\)-algebra \(\mathcal{R}\) containing all measurable subsets of the reals.
The [[borel-sigma-algebra|Borel \(\sigma\)-Algebra]] on the real numbers \(\mathcal{R}\) is the \(\sigma\)-algebra generated by the open sets (or equivalently the closed sets) of \(\mathbb{R}\).
One the measurable space \((\mathbb{R},\mathcal{R})\) we can define measures by first giving a Stieltjes measure function which is a function \(F:\mathbb{R} \to \mathbb{R}\) that is: 1. \(F(b) \geq F(a)\) for all \(b > a\) (non-decreasing); 2. \(\lim_{y \downarrow x}F(y) = F(x)\) (right-continuous).
Associated with each Stieltjes measure function \(F\) there is a unique measure \(\mu\) on \((\mathbb{R}, \mathcal{R})\) with \[ \mu((a,b])=F(b) - F(a). \] When \(F(x)=x\) the resulting measure is called the Lebesgue measure.