ABSTRACT INTEGRATION:


Abstract integration relies on a measurable space, \((\Omega,\mathcal F, \mu)\) and an \(\mathcal F\)-measurable function, \(f: \Omega \to \mathbb R,\) which in probability theory is a random variable.

The integral

\[\int_A f \;\mathrm d\mu\]

is the integral over a measurable set \(A\) of an \(\mathcal F\)-measurable function, \(f\) with respect to a measure \(\mu.\)

In the specific case when \((\Omega,\mathcal F, \mu)= (\mathbb R,\mathcal B(\mathbb R), \lambda),\) i.e. \(f: \mathbb R \to \mathbb R\) is a Borel-measurable function with the Lebesgue measure \(\lambda,\) then \(\int f\;\mathrm d \lambda\) is called the Lebesgue integral. Another specific case is the expectation of a random variable \(X:\Omega \to \mathbb R\) on a probability space is defined as \(\int X \; \mathrm d\mathbb P.\)

If the integral occurs over the entire sample space it will be implicit in the notation:

\[\int_\Omega f \;\mathrm d\mu= \int f \;\mathrm d\mu\]

Simple functions assume finitely many values in their image, and can be written as

\[f(\omega)= \sum_{i=1}^n a_i \mathbb I_{A_i}(\omega), \quad \forall \omega \in \Omega\]

where \(a_i \geq 0, \forall i \in \{1,2,3, \dots, n\},\) and \(A_i \in \mathcal F, \forall i.\)

Given the finite values in the image of the simple function \(f,\) corresponding to each of \(n\) disjoint subsets of \(\Omega,\) such that \(\bigcup_{i}A_i=\Omega,\) a simple function can indeed be written as:

\[f(\omega)=a_1 \mathbb I_{A_i}(\omega) + a_2 \mathbb I_{A_2}(\omega)+\cdots +a_n\mathbb I_{A_n}(\omega) =\sum_{i=1}^n a_i \mathbb I_{A_i}(\omega).\]

indicating that \(a_i\) will the only value in the sum if it matches the subset \(A_i\) according to the \(i\) indexation, akin to a for-loop call in programming: only the \(a_i\) corresponding to \(\mathbb I_{A_i} (\omega)=1,\) i.e. when the outcome is in the measurable set \(A_i,\) is ultimately represented.

The definition of the abstract integral of non-negative functions \(f: \Omega \to \mathbb R_+\) is defined as the supremum of collection of simple functions that approximate \(f\) from below:

Given a collection of simple functions \(q: \Omega \to \mathbb R_+\) denoted \(S(f)=\{s_1, s_2, \dots, s_N\},\) for some \(N \in \mathbb N,\) such that for all \(\omega \in \Omega\) it fulfills \(q(\omega)\leq f(\omega),\) the abstract integral is defined as

\[\int_\Omega f \;\mathrm d\mu=\sup_{q \in S(f)} \int q \;\mathrm d\mu=\sup\left(\sum_{q\in S(f)} q \; \mu (\text{preim}\left(\{q\})\right) \right)\]

There is still need to a practical method to approximating or computing this integral. The following is such a method:

\[\int_\Omega f \;\mathrm d\mu = \lim_{n \to \infty} \int f_n \mathrm d\mu\] where

\[f_n(x) = \sum_{k=0}^{n2^n-1}\frac{k}{2^n}\; \chi_{\frac{k}{2^n}\leq f(x)<\frac{k+1}{2^n}}\; +\; n\,\chi_{f(x) \geq n}\]

and letting \(n \to \infty.\)

The range or codomain of the function \(f\) is partitioned fixing a maximum value \(n\) from \(k=0\) to \(k = n\) at intervals \(\frac{k}{2^n}\) in order to define a sequence of simple functions \(f_n\) with value \(n\) if \(f(\omega)\) is equal or greater than \(n,\) and otherwise with the lowest value for that interval, \(n-\frac{1}{2^n}.\)


Coding a concept helps, so I tried doing so for this post, illustrating the Lebesgue integral of \(y = x^2\) between \([0,1]\). The code is here, and the output looks like this: