Putting letters instead of numbers on the six sides of a die does not change the codomain of the measure from $\mathbb R\cup\{\,+\infty,-\infty\,\}$ to something else. In fact, whether you use numbers or letters on the face of the die, the codomain of the measure is the set $[0,1]=\{\,x : 0\le x\le 1\,\}.$ If the letters $\text{A, B, C, D, E, F}$ have equal probabilities, then the measure assigns to the set (for example) $\{\,\text{A, B, C, D}\,\}$ the measure $2/3.$ That number, $2/3,$ is a member of the set $[0,1].$ That set remains the codomain of the measure.
Probably you meant the codomain of the random variable rather than the codomain of the measure.
It is not unusual for the codomain of a random variable to be a space of vectors rather than $\mathbb R.$ A vector-valued random variable may have an expected value in $\mathbb R^n.$
Expected values may be in any space in which it makes sense to take convex combinations.
Michael Hardy
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2026-06-15T23:37:17Z (3 months ago)