Because that’s a requirement of the approach—once it no longer holds true, we no longer increase W.
Maybe this is a better way of phrasing it: W is the space of all utility-valued random variables that have the same value as some constant (by whatever means we establish that).
Then I get linear closure by fiat or assumption: if X=c and Y=d, then X+rY=c+rd, for c, d and r constants (and overloading the = sign to mean “<= and >=”).
But my previous post was slightly incorrect—it didn’t consider infinite expectations. I will rework that a bit.
Because that’s a requirement of the approach—once it no longer holds true, we no longer increase W.
Maybe this is a better way of phrasing it: W is the space of all utility-valued random variables that have the same value as some constant (by whatever means we establish that).
Then I get linear closure by fiat or assumption: if X=c and Y=d, then X+rY=c+rd, for c, d and r constants (and overloading the = sign to mean “<= and >=”).
But my previous post was slightly incorrect—it didn’t consider infinite expectations. I will rework that a bit.