I'm not well-versed in analysis. I want to OR together an infinite number of fuzzy truths of infinitesimal significance. I mean the following:
I have an arbitrary function $t(v)$ with a range on $[0,1]$ and a domain on, say, $[0,\ell]$. There are many definitions of the fuzzy OR operator (co-norms), and I'm interested in many of them. Here are some examples: $a \lor b = \max(a, b)$; $a \lor b = a + b - ab$; $a \lor b = (a + b) / (1 + ab)$; $a \lor b = \min(a + b, 1)$. Some have awkward conditions, like $a \lor b = \max(a, b)$ if $a + b < 1$ else $1$.$
The result should be at least as true as the truest point. That is, if $t(v)$ is non-zero at only one point, the truth at that point is the result. The longer $t(v)$ is non-zero, the truer it is. That is, the OR-integral of a constant approaches $1$ as $\ell$ increases. I don't know how fast it should approach $1$. Perhaps, as $\ell$ approaches $\infty$, the or-integral of a constant approaches $1$ (I think so, but I'm not sure). How the result depends on $\ell$, I don't know. It must depend on the norm somehow. (E.g., for the $\max$ definition of OR, not at all---the result would remain $\max\left(t(v)\right)$.)
I can think of three approaches. First: by analogy to the product integral: transform the integrand into something that can be Riemann-integrated, then transform it back after integration. @Karl has shown how to do this with the for $a + b - ab$: $\rm{OR}_0^\ell\, t(v)\, dv = 1−\prod_0^\ell\, (1−t(v))\, dv$. It's not obvious to me how to do something like that for other norms.
Second: numerically, or by analogy to ORing a finite number of discrete truths. (This may be better after all, since I want to solve such problems numerically anyway.) I mean: If I sample $t(v)$ at many points $(v,t)$, I just OR all the $t$ together. This seems highly dependent on the number of $t$. I imagine that, for most norms and a lot of non-zero $t$, the result quickly approaches $1$. I fear that, because of this, my results would almost always be $1$ or $0$, which would tell me nothing---it would throw away all the information in $t(v)$.
So, I'm thinking of an analogy to Riemann integration. There, an infinite number of values are multiplied by an infinitesimal, reducing the significance of each. Maybe I could reduce the significance of each $t$ under ORing in proportion to the number of $t$ being considered, the number of sample points. (And, perhaps, take it to the limit of an infinite number of sample points of infinitesimal significance.)
The number $0$ is insignificant under ORing---ORing one constant and a million zeros results in the constant, for any norm, I think. So, if I reduce all of the $t$ to nearly zero (in proportion to their population), and OR them all together, I might be on the right track. But how exactly should I do this---by multiplying each, or taking each to a power, or by some other means---and how to do so without an underflow error if it's numerical? And, how to do this without negating the effect that more non-zero truths make the result truer?
Third: what is wrong with the following?:
for the norm $a \lor b = a + b - ab$:
$$\rm{OR}_0^\ell\, t(v)\, dv = \int_0^\ell\, t(v)\, dv - \prod_0^\ell\, t(v)\, dv$$
where, by "$\prod$" I mean the product integral.