If $F_X, F_Y$ agree for all $x \in \mathbb{R}$, Do their distributions $\mu_X, \mu_Y$ agree on $\mathcal{B}$? Let random variables X and Y be defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume their distribution functions $F_X$ and $F_Y$ agree for all $x \in \mathbb{R}$. How would I conclude that that their distributions $\mu_X$ and $\mu_Y$ agree on $\mathcal{B}$?
Would the same hold if the assumption was that $F_X$ and $F_Y$ agreed for all $x \in \mathbb{Q}$?
 A: Observation $1.$ Let $\mathbf P_1, ~\mathbf P_2$ be two probability measures on $(\Omega, \mathcal F). $ Let $\mathcal P$ be a $\pi$-system such that $$\mathbf P_1(A) =\mathbf P_2(A), ~~~\forall A\in \mathcal P. \tag 1$$ Then $$\mathbf P_1\left(A^\prime\right) =\mathbf P_2\left(A^\prime\right), ~~~\forall A^\prime\in \sigma(\mathcal P) . \tag 2$$
Proof: It is easy to check that $\mathcal L:=\{A\in \mathcal F: \mathbf P_1(A) =\mathbf P_2(A)\}$ is a $\lambda$-system. By $(1),~\mathcal P\subset \mathcal L$ whence by Dynkin's theorem, $\sigma(\mathcal P) \subset \mathcal L. $
Observation $2.$ If $\mathbf P_1, ~\mathbf P_2$ are probability measures on $(\mathbb R, \mathcal B) $ such that $$\mathrm F_1(x) = \mathrm F_2(x), ~~\forall x\in \mathbb R, \tag 3$$ then $$\mathbf P_1\equiv \mathbf P_2\tag 4$$ on $\mathcal B. $
Proof. Take $\mathcal P:= \{(-\infty,x]: x\in \mathbb R\};$ it is a $\pi$-system. Also $\sigma(\mathcal P) =\mathcal B. $ By Observation $1,  ~(3)\implies (4).$
Theorem $1.$ Let $X, ~Y$ be random variables on $(\Omega, \mathcal{F}, \mathbf{P}) .$ Then $$ \mu_X\equiv \mu_Y \iff \mathrm F_X(x) =\mathrm F_Y(x) ~~\forall x\in \mathbb R. \tag 5$$
Proof. $(\Leftarrow) $ Use Observation $2.$
$(\Rightarrow) $ Take the sets $(-\infty, x].$
As for whether it is true for $x\in \mathbb Q, $ check whether $\mathcal P^\prime := \{(-\infty,x]: x\in \mathbb Q\}$ is a $\pi$-system and whether or not $\sigma(\mathcal P^\prime) =\mathcal B. $ Check if it is possible to use the same approach above or not.

Reference:
$[\rm I]$ A Probability Path, Sidney I. Resnick, Birkhäuser, $1999$; chapter $2, $  pp. $37-38.$
