Please write proofs and full explanations for the following question. Really appreciate your help!!

Note: the MCS --- http://www.cs.cornell.edu/courses/cs2800/2017sp/handouts/mcs.pdf

When constructing probability spaces with finite sample spaces, it is convenient to only specify the probability of simple events: those containing only a single outcome. in fact, this is how MCS defines a probability space. Show that this is justified: given a function P: S rightarrow R satisfying for all s Element S', P(s) greaterthanorequalto 0, and sigma_s Element S P(s) = 1, show that the function P: 2^s rightarrow R given by P_r(E) = sigma_s Element E P(s) satisfies the axioms for a probability measure. Show that Pr is unique, in the sense that any probability measure Pr' satisfying Pr'({s}) = P(s) must be equal to Pr. Another simple way to construct a probability space with a finite sample space is to use an equip rob-able measure: define P_r(E) = |E|/|S|. Show that this definition of Pr satisfies the axioms for a probability measure. When constructing probability spaces with finite sample spaces, it is convenient to only specify the probability of simple events: those containing only a single outcome. in fact, this is how MCS defines a probability space. Show that this is justified: given a function P: S rightarrow R satisfying for all s Element S', P(s) greaterthanorequalto 0, and sigma_s Element S P(s) = 1, show that the function P: 2^s rightarrow R given by P_r(E) = sigma_s Element E P(s) satisfies the axioms for a probability measure. Show that Pr is unique, in the sense that any probability measure Pr' satisfying Pr'({s}) = P(s) must be equal to Pr. Another simple way to construct a probability space with a finite sample space is to use an equip rob-able measure: define P_r(E) = |E|/|S|. Show that this definition of Pr satisfies the axioms for a probability measure