Why is the measure of this set 0?

Question

Williams has the following note in his book Probability with Martingales:

Lemmma 5.2b simply states that

I don't see why $\mu(\{L\neq U\})=0$. I tried doing a proof by contradiction (If possible, let $\mu(\{L\neq U\})>0$, but that didn't lead my anywhere. Any hints?

Answer 1

By virtue of choice, we know that $L \leq U$ and $$\mu(U-L) = I-I = 0.$$ This means that $U-L$ is a non-negative random variable with expectation $0$, hence $U-L=0$ $\mu$-almost everywhere. Indeed: As $U \geq L$, we know that

$$\mu(\{U \neq L\}) = \mu(\{U>L\})=\mu(\{U-L>0\}).$$

Applying the lemma yields $\mu(\{U \neq L\})=0$.