Could monotone class theorem be applied to the proof of inequality (of probability) I have 4 independent random variables: $X_1,X_2,Y_1,Y_2$. And I know for any measurable sets $S_1 \in \mathcal{F}_1, S_2 \in \mathcal{F}_2$, I have
$$\Pr(X_1 \in S_1) \leq c\Pr(Y_1 \in S_1)$$
$$\Pr(X_2 \in S_2) \leq c\Pr(Y_2 \in S_2)$$
for some constant $c>1$.
By independence, I deduce that
$$\Pr((X_1,X_2) \in S_1 \times S_2) \leq c^2\Pr((Y_1,Y_2) \in S_1 \times S_2)$$
Now, the question is: does this property hold for all sets in the product $\sigma$-algebra $\sigma(\mathcal{F}_1 \times \mathcal{F}_2)$?
We cannot use the monotone class theorem here, for this property does not hold for $\lambda$-system.
 A: Use that the set $S$ of finite disjoint unions of "rectangles" forms an algebra and use the monotone class theorem (http://en.m.wikipedia.org/wiki/Monotone_class_theorem) as opposed to Dynkins $\pi\lambda$ theorem about which you seem to be thinking. 
EDIT: Second possibility: It is (at least for finite (positive) measures) not hard to see $(\mu +\nu)\otimes \gamma =\mu \otimes \gamma +\nu \otimes \gamma$, wher $\otimes$ denotes the product measure. 
For a finite signed measure $\nu =\mu_1 -\mu_2$ with finite positive measures, it is thus easy to see that $\nu \otimes \gamma :=\mu_1\otimes \gamma -\mu_2\otimes\gamma$ is well-defined, independent of the decomposition chosen for $\nu$. 
Now, if $\mu\leq \nu$ and $\gamma$ is a finite positive measure, we get $0\leq (\nu -\mu)\otimes \gamma$, which by linearity yields $\mu \otimes \gamma \leq \nu\otimes \gamma$. 
A slight generalisation also shows $\gamma \otimes \mu \leq \gamma \otimes \nu$. 
Now your assumptions are
$$
\mu_i \leq c \nu_i
$$
for $i \in \{1,2\}$, where the $\mu_i,\nu_i$ are the apropriate image measures. Now the above considerations imply your claim. 
