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I wonder what benefits or purposes can be obtained, through studying measures completely in terms of some or all measurable mappings that can induce the measures on their codomains?

Since different measurable mappings defined from the same measure space to the same codomain can induce the same measures on their codomains, representing measures by measurable mappings actually induce more complication to consider.

For example,

  • In many probability examples/exercises, the questions being asked are entirely about the probability measures, but they are phrased completely in terms of random variables that induce the probability measures on their codomains.
  • Coupling, as I understand and if I understand correctly, study the relation between two probability measures, in terms of any pair of two random variables that have the two probability measures as their induced ones respectively.

Thanks!

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