Somewhat like Computer Science, Probability Theory has the nasty habit of renaming well-known concepts from Functional Analysis to make them conform with its somewhat peculiar world view. However, Probability isn't nearly as bad; usually only ONE term is invented, and there is an innate order in the new names. In the hope of aiding the student of either field who is moving to the other (and especially to help myself remember translations I already have), I present a dictionary of equivalent terms.

sample space == σ algebra
The sample space is just the set of subsets of our space Ω for which P(.) is defined. As such, it's just a σ algebra.
probability == finite measure
Yup, it's (almost) true. A probability is just a measure P on Ω for which P(Ω)=1. If we have any finite measure μ on Ω, we can normalize it by defining P(A)=μ(A)/μ(Ω), getting a probability measure. So there's really no difference.

If μ(Ω)=∞, we cannot normalise. In particular, we cannot take Lebesgue measure on the real line and get a uniform distribution on the real numbers; the two envelope paradox shows no such creature exists.

event == measurable set
An event is something for which the measure P is defined; that's known as a measurable set.
random variable == measurable function
expectation == integral
It's easy:
E(X) = ∫ΩX(ω) dP(ω)
/msg ariels