Are these two definitions of independence of random variables equivalent? I was taught that two random variables $\xi$ and $\eta$ are independent if and only if:
$$\forall a\in\mathbb{R},\forall b\in\mathbb{R}: \mathbb{P}(\xi<a,\eta<b)=\mathbb{P}(\xi<a)\mathbb{P}(\eta<b) \quad \quad (1)$$
At the same time I've seen alternative definition in some other books:
$$\forall A\in \mathscr{B}(\mathbb{R}),\forall B\in\mathscr{B}(\mathbb{R}): \mathbb{P}(\xi\in A,\eta\in B)=\mathbb{P}(\xi\in A)\mathbb{P}(\eta\in B) \quad \quad (2)$$
I've tried to prove that these definitions are equivalent. It was quite easy to prove that $(2)$ implies the $(1)$. But I couldn't prove the converse implication. I planned to show that the set: $$L(A)=\{B\in\mathscr{B}(\mathbb{R}):\mathbb{P}(\xi\in A,\eta\in B)=\mathbb{P}(\xi\in A)\mathbb{P}(\eta\in B)\}$$
is a $\sigma$-algebra for any $A\in\mathscr{B}(\mathbb{R})$ and that it contains all sets of the form $(-\infty,a)$ with $a\in\mathbb{R}$. Which would show that $L(A)=\mathscr{B}(\mathbb{R})$ for any $A\in\mathscr{B}(\mathbb{R})$ which would show the converse. But I had problems with showing that $L(A)$ really is a $\sigma$-algebra. I've managed to show that it is closed under countable disjoint unions, relative complementaion $A\setminus B$
when $B\subset A$ and that it is a monotone class. But I couldn't prove that it is closed under intersection or full relative complementation. So is it really the case that these defintions are equivalent, and if this is the case could you give a hint how could I proceed in my proof or prove it some other way ?
 A: First note that the intersection of two intervals of the form $(-\infty,x)$ is again of the same form. Write $\mathcal{I}$ for the family of such intervals; they form a $\pi$-system. As you have observed, the family of Borel sets $B$ such that
$$\mathbb{P}(\xi\in A,\eta\in B)=\mathbb{P}(\xi\in A)\mathbb{P}(\eta\in B)$$
for all $A\in\mathcal{I}$ are a Dynkin sytem that, by assumption, includes all elements of the $\pi$-system $\mathcal{I}$. By the $\pi-\lambda$-theorem, this family includes the $\sigma$-algebra generated by $\mathcal{I}$, which includes all Borel sets. You can then repeat the exercise, for the family of Borel sets $A$ such that $$\mathbb{P}(\xi\in A,\eta\in B)=\mathbb{P}(\xi\in A)\mathbb{P}(\eta\in B)$$
for all Borel sets $B$. From what we have shown before, it includes the $\pi$-system
$\mathcal{I}$ and forms a Dynkin system. Using again the $\pi-\lambda$-theorem, we can conclude that we can allow for $A$ all Borel sets. So it works for $A$ and $B$ being arbitrary Borel sets.
