I was trying to understand the difference between the concepts of probability distribution and probability space for what concerns the assignment of probability over the sample space. Wikipedia gives the following definition for probability distribution:

A probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

Which actually is the same definition which it gives also for a probability space, even if for this last one more formally introduces the concepts of a sample space $Ω$, an event space *F$ (a σ-algebra) and a probability function giving measures of probability for the events. Also, I would like to point out that a probability function is indeed one of the possible ways to uniquely define a probability distribution.

I would like to know if there is some difference between the two terms, if the difference is more of a contextual convention nuance, or if this is the actual formality and indeed which in case is the actual difference.