$F(x)=m(A\cap(-\infty,x))$ for $x\in\mathbb{R}$ then $F$ is continuous on $\mathbb{R}$.

Question

If $A\subseteq\mathbb{R}$ is measurable with finite Lebesgue measure and $F(x)=m(A\cap(-\infty,x))$ for $x\in\mathbb{R}$ then prove that $F$ is continuous on $\mathbb{R}$.

I actually am quite lost here. Can anyone help me with this?

Answer 1

For any increasing sequence $(x_n)$ with $x_n\rightarrow x $ we have $$ |F(x_n)-F(x)|=|m(A\cap(x_n,x))|\leq |x_n-x|\rightarrow 0 $$ and so $F(x_n)\rightarrow F(x)$, proving continuity.