I am reading Billingsley (1968).Convergence of Probability Measures. In page 236, he mentioned four types of bases for weak topology of probability measure.
And when he proved "find within (5) a set of the form (6)."
He wrote that
$\int fdQ<\epsilon+\frac{1}{k}\sum_{i}Q\left(F_{i}\right)$
I think this statement is not correct since $F_{i}=\left\{ x\in X:\frac{i}{k}\le f\left(x\right)\right\} $, we have
$\int fdQ \geq \frac{1}{k}\sum_{i}i*Q\left(F_{i}\right)$
but since $\frac{1}{k}<\epsilon$, there is no guarantee that
$\frac{1}{k}\sum_{i}iQ\left(F_{i}\right)<\epsilon+\frac{1}{k}\sum_{i}Q\left(F_{i}\right)$
In this case, I don't think the proof work. Am I wrong?