If I may ask, why didn’t you use the following (simpler imo) example: pmf_A(0) = 1 pmf_A(1) = 0 pmf_B(0) = 0 pmf_B(1) = 1
With that approach one can argue that the two PMFs have different ranges[1], and get rabbit-holed into a discussion of e.g. “is a uniform distribution from 0 to 1 with a range of −10 to 10 the same or different than a uniform distribution from 0 to 1 with a range of 0 to 1”.
This approach is more complex, but sidesteps that.
With that approach one can argue that the two PMFs have different ranges[1], and get rabbit-holed into a discussion of e.g. “is a uniform distribution from 0 to 1 with a range of −10 to 10 the same or different than a uniform distribution from 0 to 1 with a range of 0 to 1”.
This approach is more complex, but sidesteps that.
(Also see https://www.lesswrong.com/posts/mERNQwDNTtqsXbSng/you-can-tell-a-drawing-from-a-painting?commentId=ySCpKgJ8WmN7BFJjN)
What about
pmfA(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩12if x=012if x=20otherwise
pmfB(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩13if x=013if x=113if x=20otherwise ?
Both functions’ support has the same minimum (0) and maximum (2).