**Measure and Probability (Proofs)**

**Proposition 1.** The measure of the empty set is always 0. In other words:

**Proof of Proposition 1.** Since

the empty set is disjoint from itself. So, rule 2 of measures implies:

On the other hand,

implies:

Combining Equations (1) and (2) we get:

Subtract from both sides of Equation (3) to get:

So Proposition 1 is proven.

**Proposition 2.** Suppose that *A* and *B* are subsets of *S *and that *A* is contained in *B* then

**Proof of Proposition 2.** Write *B* as the disjoint union:

Rule 2 of measures then implies:

Rule 1 of measures implies:

So:

So Proposition 2 is proven.

**Proposition 3.** If , then .

**Proof of Proposition 3. *** A* and *S* are subsets of *S*, so Proposition 2 implies . Rule 1 of measures indicates that .