What assumptions are needed to get two integrals close to each other?

Question

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions do I have to make on $A,B,C$ to get this to happen?

Answer 1

I'm not sure if this is what you're looking for, but if $B$ is bounded away from zero (i.e., if there exists $M$ such that $\frac{1}{|B(x)|} \leq M$ for all $x$, or almost all $x$), then we could say the following:

$$ \left| \int A - \int \frac{C}{B} \right| = \left| \int \left( A - \frac{C}{B} \right) \right| \leq \int \left| A - \frac{C}{B} \right| = \int \frac{1}{|B|} \left| AB - C \right| \leq M \int \left| AB - C \right| \leq M \epsilon $$

So we would have that $\int A$ is within $M \epsilon$ of $\int \frac{C}{B}$.