Divide the interval $[0,1]$ in $n$ subintervals of length $\frac{1}{n}$. Assign a color to each subintervals alternating black subintervals to white subintervals.
Call $B_n$ the union of all black subintervals and $W_n$ the union of all white subintevals.
Let $J:[0,1]\times [0,1]\to \mathbb R$ a smooth, symmetric function such that $\int_0^1J(x, y)\,dy\equiv 1$.
Is that true that
$$\sup_{x\in [0,1]}\left|\int_0^1J(x, y)\chi_{B_n}(y)\,dy-\frac{1}{2}\right|\xrightarrow[n\to +\infty]{}0?$$
In other words is that true that $$\sup_{x\in [0,1]}\left|\int_{B_n}J(x, y)\,dy-\frac{1}{2}\right|\xrightarrow[n\to +\infty]{}0?$$
Could someone explain me why?
I found that step on a paper but it is not clear for me (they say that it follows by the continuity of the function $J$).
Thank you!