apply the law of total expectation

Question

I'm a little bit confused about applying the law of total expectation. Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, which one of the following equalities is correct?

$1.$

\begin{align} & E_2[E_3[\max\{v_2,v_3\}\mid v_1<\max\{v_2,v_3\}]] \\[8pt] = {} &\Pr(v_2=\max\{v_2,v_3\})E_2[v_2\mid v_1 < \max\{ v_2, v_3\}, v_2=\max\{v_2,v_3\}] \\[8pt] & {} +\Pr(v_3=\max\{v_2,v_3\})E_3[v_3\mid v_1< \max\{v_2,v_3\}, v_3=\max\{v_2,v_3\}] \end{align}

$2.$

\begin{align} & E_2[E_3[\max\{v_2,v_3\}\mid v_1<\max\{v_2,v_3\}]] \\[8pt] = {} & \Pr(v_2=\max\{v_2,v_3\}\mid v_1<\max\{v_2,v_3\}) E_2[v_2\mid v_1 < \max\{v_2,v_3\},v_2 =\max\{v_2,v_3\}] \\[8pt] & {} + \Pr(v_3 = \max\{v_2,v_3\}\mid v_1 < \max\{v_2,v_3\}) E_3[v_3\mid v_1 < \max\{v_2,v_3\}, v_3=\max\{v_2,v_3\}] \end{align}

Answer 1

For continuous random variables, the probability of a tie is immeasurably small.   So we can say: $$V_1<\max\{V_2, V_3\} \iff V_2=\max\{V_1,V_2, V_3\} \cup V_3=\max\{V_1, V_2, V_3\}\quad\text{a.s.}$$

Since the events in this union are disjoint (almost surely), then they partition the conditioned space:

$$\begin{align}\mathsf E[\max\{V_2,V_3\}\mid V_1<\max\{V_2,V_3\}] & = {\mathsf E[V_2\mid V_2=\max\{V_1,V_2,V_3\}]\;\mathsf P(V_2=\max\{V_1,V_2,V_3\}) + \\ \; \mathsf E[V_3\mid V_3=\max\{V_1,V_2,V_3\}]\;\mathsf P(V_3=\max\{V_1,V_2,V_3\})}\end{align}$$