Proof that neither “almost none” nor “almost all” functions which are Lebesgue measurable are non-integrable According to answer in (https://mathoverflow.net/a/28114/87856) and (https://www.ams.org/journals/bull/2005-42-03/S0273-0979-05-01060-8/S0273-0979-05-01060-8.pdf) "almost all" can be defined without a measure on the set of measurable functions.
Here is what I have so far:
In order for a function to be Lebesgue integrable, it must be finite almost everywhere and the set of points at which it is infinite must have measure zero. The fact that the function is made up of pseudo-random, non-uniform points does not necessarily mean that it is not Lebesgue integrable. A Lebesgue-measurable function is Lebesgue-integrable if the integral of the absolute value of the function over the set is finite. To determine if a function is Lebesgue-integrable, one would need to calculate the integral of the absolute value of the function over the set and determine if it is finite.
I have many examples on both sides (Lebesgue measurable functions that are integrable vs. Lebesgue measurable functions that are non-integrable and think it would not be possible to say if integrable or non-integrable functions are the majority, etc.
I believe there are many Lebesgue measurable functions that are integrable and there are many that are not. The set of Lebesgue measurable functions is uncountable as is the set of integrable or non-integrable ones. So, unless a specific set of these is defined, we would not be able to tell which is more than the other (both cardinally or the population-wise).
 A: As requested in a comment, below is my 21 Jan 2021 answer to  Size of a function space, a question that was deleted by "Community" approximately 1 year ago. The only change I've made is to include a reference to my 18 Sep 2020 answer that briefly discusses porous and $\sigma$-porous sets. This doesn't answer the question asked, but it does point the way toward topics relevant to various ways in which the question could be answered.
Fractal Dimension Notions
All of the more common fractal dimension notions (box/Minkowski, packing, Hausdorff) have gauge function versions that allow for measuring the size of subsets of infinite dimensional metric or normed spaces. For an introduction to this and a guide to some of the literature, see Mark C. McClure's 1994 Ph.D. dissertation Fractal Measures on Infinite Dimensional Sets and Joseph Michael Strus's 1994 Ph.D. dissertation Metric Entropies of Various Function Spaces.
Kolmogorov Entropy
Russian mathematicians have done an extensive amount of work on various measures of the sizes of function spaces (some of this is discussed in McClure's dissertation), and a well-known survey paper that has been cited a huge number of times (and thus, google searching for citations of it will lead to a large number of subsequent papers) is $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces by Kolmogorov/Tikhomirov (Russian original published in 1959, AMS English translation published in 1961).
Baire Category and Porosity
The smallness notion of a "first category set" (also called a "meager set") from the Baire category theorem has been extensively applied in function spaces since the late 1920s. In the last 30 or so years, stronger notions of "first category set" defined as various notions of porous and $\sigma$-porous sets (see also this answer), including the use of gauge functions for additional range and precision of application, have been applied to function spaces. As an analogy, one can think of various $\sigma$-porous notions as refining "first category", similar to how various fractal dimensions refine "measure zero".
Prevalence
Also in the past 30 or so years, the notion of prevalence (also called Haar null sets) has been widely applied in function spaces. A well-known survey paper that has been cited a huge number of times (and thus, google searching for citations of it will lead to a large number of subsequent papers) is Prevalence: A translation-invariant "almost every" on infinite-dimensional spaces by Hunt/Sauer/Yorke.
Other Notions
Other notions include $\Gamma$-null sets, $\Gamma_n$-null sets, cube null sets, Aronszajn null sets, Gauss null sets, etc. that you can find discussed in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces by Lindenstrauss/Preiss/Tišer and in the papers collected together in Proceedings of the International Workshop on Small Sets in Analysis edited by Matouskova/Reich/Zaslavski (these papers were also published in Issues 3-5 of Volume 2005 of the journal Abstract and Applied Analysis).
