You could use hyperreal numbers. They behave pretty similarly to reals, and have reals as a subset. Also, if you multiply any hyperreal number besides zero by a real number, you get something isomorphic to the reals, so you can multiply by infinity and it still will work the same.
I'm not a big fan of the continuity axiom. Also, if you allow for hyperreal probabilities, you can still get it to work.
Also, if you multiply any hyperreal number besides zero by a real number, you get something isomorphic to the reals,
True
so you can multiply by infinity and it still will work the same.
Only if you have a way to describe infinity in terms of a real number.
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