How would one use the pi-lambda theorem in determining that a CDF completely defines the distribution of a random variable? The pi-lamda theorem states that if $Q$ is a pi-system contained in a lambda system $L$, then the sigma-algebra generated by $Q$ is also contained in $L$. We can use this to show that the CDF completely determines the distribution of a random variable, but I do not understand the connection. 
 A: For a (real, Borel-measurable) random variable $X$ on a space with probability measure $P$, we define the distribution function as $F_X(x)=P(X\le x)$. Let $P_X$ be the distribution. This is a measure on $\mathbb R$. 
By definition $P_X(A)=P(X^{-1}(A))$. Note if we know $F_X(x)$, we know $P_X(A)$ when $A$ is of the form $(-\infty, x]$. Sets of this form are a $\pi$-system, so the $\sigma$ algebra they generate is equal to the $\lambda$ system they generate. Using the complement property of $\lambda$ systems, we see the $\lambda$ system generated by sets of this form is equal to the $\lambda$ system generated by sets of the form $(a,b]$. Taking countable disjoint unions shows we may permit $b=\infty$ in our generating set without changing the generated $\lambda$-algebra. Then taking complements again shows that the $\lambda$ system these sets generate is is equal to the $\lambda$ system generated by sets of the form $(r,s)$. These sets form a $\pi$-system, so the $\lambda$ system they generate is equal to the $\sigma$-algebra they generate, which is the collection of Borel sets, as desired. 
This proof is a little roundabout. There are more elementary ways of reaching this conclusion. See, for example, the first chapter in Folland's Real Analysis. 
