Abyss, on 26 June 2023 - 04:36 PM, said:
Azath Vitr (D, on 25 June 2023 - 07:06 PM, said:
Math/measure theoretic probability textbook marketed for 'self-study' is full of errors not included in the errata and skips lots of steps while getting what 'work' it does show wrong.
For example it uses integration by parts in a proof but has 'v' where it should say 'dv/dx' and omits the algebra that would demonstrate its error... it would be easy to follow along if it showed more of the steps, but checking this shit is tedious.
It doesn't help that the usual formula for integration by parts is needlessly confusing: integral u dv = vu - integral v du instead of the equivalent formula u (dv/dx) dx = integral v (du/dx) dx which is easier to remember and re-derive (because by the product rule d(uv)/dx = u(dv/dx)+v(du/dx) so taking antiderivative of both sides we get uv = integral (u (dv/dx) + v (du/dx)) dx which by linearity of the integral = integral u (dv/dx) dx + integral v (du/dx) dx
AV are you high as fuck or did you get hacked?
I do like listening to mathematicians who are tripping on LSD and talking about cutting edge geometries.
Simple as it is by comparison, calculus is ravishingly
beautiful---and sublime.
But the standard integration by parts formula is ugly and obfuscatory. Whereas the expression using dv/dx / du/dx is
claire, luxe, et volupté.
Of course Riemann, probably best known in the popular imagination for non-Euclidean geometry (eventually leading to the non-Euclidean geometry of General Relativity), is also who the standard elementary calculus integral is named after....
In measure theoretic probability, the Lebesgue integral beautifully turns the Riemann integral on its head and inside out---instead of infinitely shrinking rectangles rising from the x axis down to vertical strips to get arbitrarily close to the area under the curve, it assigns a length-like 'measure' to the inverse image of the y values (so the x value can discontinuously jump around infinitely often, instead of being a continuous rectangle) and if certain conditions are met (also required for Riemann integration) then gets arbitrarily close to the area under the curve with an enumerably infinite sequence of weighted sums of finite sequences.
(Measure theory is necessary for a rigorous understanding of modern probability theory.)
This post has been edited by Azath Vitr (D'ivers: 26 June 2023 - 09:13 PM