I’m not sure if clarifying this is most useful for the purpose of understanding this post specifically, but for what it’s worth: A measurable space is a set together with a set of subsets that are called “measurable”. Those measurable sets are the sets to which we can then assign probabilities once we have a probability measure P (which in the post we assume to be derived from a density p, see my other comment under your original comment).
“the function X is constant,” you mean its just one outcome like a die that always lands on one side?
I think that’s what the commenter you replied to means, yes. (They don’t seem to be active anymore)
what makes a function measurable?
This is another technicality that might not be too useful to think about for the purpose of this post. A function is measurable if the preimages of all measurable sets are measurable. I.e.: f:X→Z, for two measurable spaces X and Z, is measurable, if f−1(A)⊆X is measurable for all measurable A⊆Z. For practical purposes, you can think of continuous functions or, in the discrete case, just any functions.
I’m not sure if clarifying this is most useful for the purpose of understanding this post specifically, but for what it’s worth: A measurable space is a set together with a set of subsets that are called “measurable”. Those measurable sets are the sets to which we can then assign probabilities once we have a probability measure P (which in the post we assume to be derived from a density p, see my other comment under your original comment).
I think that’s what the commenter you replied to means, yes. (They don’t seem to be active anymore)
This is another technicality that might not be too useful to think about for the purpose of this post. A function is measurable if the preimages of all measurable sets are measurable. I.e.: f:X→Z, for two measurable spaces X and Z, is measurable, if f−1(A)⊆X is measurable for all measurable A⊆Z. For practical purposes, you can think of continuous functions or, in the discrete case, just any functions.