Let $(X,\mathcal{A},\mu,\phi)$ be a topological dynamical system, where $X$ be the circle represented as $\mathbb{R}/\mathbb{Z}$, $\mathcal{A}$ be the Borel $\sigma$-algebra subsets of $X$, $\mu$ Lebesgue measure and $\phi\colon X\rightarrow X$, where $$\phi(x)=x+a$$ for an arbitrary $a$.
I wish to know how to prove that the function $$g(y)=\int_{A}\chi_{A}(x+y)\,d\mu(x)$$ is continuous, where $A\subset X$ and $\chi_{A}$ is the characteriscti function of $A$.
My approach to the problem so far: I tried to deal with the classical definition of continuity, considering $y,y_0$ with close distance and tried to estimate the quantity $$\left|\int_{A}\chi_{A}(x+y)\,d\mu(x)-\int_{A}\chi_{A}(x+y_0)\,d\mu(x)\right|=\left|\mu(A\cap(A-y))-\mu(A\cap(A-y_0))\right|.$$
I cannot proceed from this point to see how this quantity is small, based on the distance of $y,y_0$. I am gratefull for any help you give me. Thanks in advance.