Probability involving Random Variables

Question

Let $A$ and $B$ be two Random Variables. If $A$ has a continuous CDF and $A$ and $B$ are independent, prove that :

$$ P(A - B = c) = 0 \ \forall c \in \mathbb{R} $$

I don't know how to proceed with this one.

Please help.
Thanks in advance.

Answer 1

Let $c\in\mathbb{R}$ be fixed and let $\left[a-b=c\right]$ denote the function $\mathbb{R}^{2}\to\mathbb{R}$ that takes value $1$ is $a-b=c$ and takes value $0$ otherwise.

Then:

$$\begin{aligned}\mathsf{P\left(A-B=c\right)} & =\iint\left[a-b=c\right]dF_{A,B}\left(a,b\right)\\ & =\iint\left[a-b=c\right]dF_{A}\left(a\right)dF_{B}\left(b\right)\\ & =\int\mathsf{P}\left(A=b+c\right)dF_{B}\left(b\right)\\ & =\int0dF_{B}\left(b\right)\\ & =0 \end{aligned} $$

The second equality applies independence of $A$ and $B$.

The fourth equality applies that $A$ has a continuous CDF.