(Reformatted Latex so the comment text editor won’t reject it.)
If you have a point M∈B, and some other point M∗ that’s an sa-measure, we might as well add M+M∗ to B. Why? Well, given some positive functional f+ (and everything we’re querying our set B with is a positive functional by Proposition 1,
f∗(M+M∗)=f∗(M)+f∗(M∗)≥f∗(M)
There’s a missing end parenthesis (to match the opening parenthesis on line 2), although it’s not completely clear where it goes:
replacing “Proposition 1,” with “Proposition 1.” or
at the end of the equation (at the end of the quote).
(Maybe there’s also something else going on with the paragraph after the quote, which continues the sentence.)
“add all the points you possibly can that don’t affect the inf(m,b)∈B(m(f)+b) value for any f“[.]
The set of minimal points is denoted Bmin[.]
Looking back at our second desideratum, it says “Our notion of a hypothesis in this setting should collapse “secretly equivalent” sets, such that any two distinct hypotheses behave differently in some relevant aspect. This will require formalizing what it means for two sets to be “meaningfully different”, finding a canonical form for an equivalence class of sets that “behave the same in all relevant ways”, and then proving some theorem that says we got everything.”
[Emphasis added.]
Some way of noticing that first ” has been opened and isn’t closed until the end of the paragraph might make the quotes section easier to parse—at the risk of making a section stick out due to complexity rather than because you want to emphasize it a lot.
(Reformatted Latex so the comment text editor won’t reject it.)
There’s a missing end parenthesis (to match the opening parenthesis on line 2), although it’s not completely clear where it goes:
replacing “Proposition 1,” with “Proposition 1.” or
at the end of the equation (at the end of the quote).
(Maybe there’s also something else going on with the paragraph after the quote, which continues the sentence.)
[Emphasis added.]
Some way of noticing that first ” has been opened and isn’t closed until the end of the paragraph might make the quotes section easier to parse—at the risk of making a section stick out due to complexity rather than because you want to emphasize it a lot.