??"entropy" of a probability measure, or something...

??say x has weight 1/2, and y and z each 1/4 ... then use 1 bit to record x, two to record y or z ... 1/2 * 1 + 1/4 * 2 + 1/4 *2 ....

= expected 3/2 bits per selection ... ???

vs x,y each 1/2 ...

1/2 * 1 + 1/2 * 1 ....

=1

1/2 1/2 1

1/4 1/4 1/4 1/4 2

1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 3

????something about... "maximizing over the ways of using probability measure x as a random-bit generator, of the average amount of information needed to record... " ... ???something about tweaking by y ... ??or something ???...

??something about ... ??"proportion of randomness of x which can be filtered through y" .... ???or something ???....

??idea that most sensible thing to attribute "entropy" to is... ??a "random variable" (or something ...) ...??especially (maybe not exclusively though??) finite-codomained, or something???... ??something about "expected length of recorded results... quantifying over possible coding schemes ... " ... ??or something ... ???...

(hmm, might i be making rota's mistake here??? ????or something ??? ....)

??what about whether there might be something about ... ???getting weighted colimits _and_ weighted limits involved here ???.... something about decategorification ... exponentiation ... and so forth ... ???....

???so then (...) what _about_ something about ... ??... ??given "alternative probability measure" (or something ... ???maybe something "riesz-ish" (??or something??? ?? ??maybe phrase "absolutely continuous" popping into my mind here ??... ??hmm, maybe i was groping for something like "radon-nikodym derivative" here?? ??? or something?? ... and so forth ...) going on here??? ...), considering entropy of "the (??...) filter yielding it (??in bayesian fashion) from the original probability measure" ?? ... or something... and so forth ... ??? something about ... ??"filter" as boolean random variable ... ??something about negation / complement filter here, and something about ... "distributing (or something ... ??"attributing" ... ??...) total entropy of question among its various answers ..." ... ???attributing entropy more fundamentally to "answered question" than to "question" ... or something ... and so forth ... ???.... ???something about ... highly restrictive filter having lowly restrictive one as negation / complement ... ???... ??hmm, so what about something about geometry (or something ...??...) of 3 probability measures in row (or something?? ...) : no-update, original, yes-update ... ?? ... and so forth ... ???...

???so what _about_ something about ... ?? "filter with continuous support but finite entropy" (??or something ... and so forth ... ???hmmm, what about something about continuous _filter_ here ??? .... or something ... and so forth ...) ?? ... ???any relationship to ... ???"conceptual foundations of quantum mechanics" (or something... and so forth ...) ?? ... ???something about "discrete spectrum vs continuous spectrum" ?? ... and so forth ... ??? ....