Does $\left\lVert a-b \right\rVert_\infty < \epsilon \implies (\int a-b)< \int \epsilon$? $a,b$ are continuous functions.
I think this is true as constants are continuous functions but I'm not sure if it holds for all cases.
Edit: Integrated over two real numbers

Only if the measure space is finite, otherwise you'll have only $\int (a-b) \le \int \varepsilon$ because both sides may be infinite.$\,$
