Lebesgue integral
NOTES ON STATISTICS, PROBABILITY and MATHEMATICS
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:
Finally, here is a systematic “construction” of the Lebesgue integral for an inverted parabola, really showing how the critical step is to partition the range (y axis) in \(N\) equally spaced divisions between the \(y\) limits of integration, defining \([y_i,y_{i+1}]\) intervals, and look for the measure of the pre-image in the \(x\) axis: the difference between the inverse function at a given chosen point, \(y^*\) (not a slab) for any given \([y_i, y_{i+1}]\) interval, \(f^{-1}(y^*_i)\), and the inverse of the next point in the y axis, \(y^*_{i+1}\). Analytically, this “strategy” relies ultimately in letting \(N\rightarrow\infty\), which explains the “pyramid” iconography so often found online, which is intuitive, but has the potential to suggest the integral as a pile of horizontal slabs. This was a big hurdle understanding the difference between the Lebesgue and the Darboux integrals. Here is an attempt at a more faithful dynamic representation (again the code is here waiting for improvements):
When we extend the partitions to \(100\) (a humble computer version of \(\infty\)) the calculated area is \(1.322926\), really close to the analytical value, and identical to the Riemann integral \(2\int_0^1 (-x^2 +1)\, dx=4/3.\)
NOTE: These are tentative notes on different topics for personal use - expect mistakes and misunderstandings.