Kobayashi, Mark & Turin's Probability, Random Processes and Statistical Analysis, 2012, states without proof:
three events, A, B, C are mutually independent when:
P[A,B]=P[A]P[B], P[B,C]=P[B]P[C], P[A,C]=P[A]P[C], P[A,B,C]=P[A]P[B]P[C]
No three of these relations necessarily imply the fourth. [my italics]
However, Wikipedia and others generally agree that mutual independence implies pairwise independence, but also without a demonstration.
What is the simplest proof that mutual independence implies pairwise independence?
Note: GC Rota wrote that probability can be understood by focusing on random variables or focusing on distributions. However, the two views should be equivalent, correct?