@WillieWong @WillieWong So are you officially done answering my question. I wonder what your "complete definition" looks like. If $\lambda(A)>0$, does the average of $f$ give the same result as the Lebesgue-Khichnine integral? When $\lamda(A)=0$ and $A$ is uncountable do countable subsets of $A$ have "zero effect" on the average of $f$ while uncountable subsets carry "greater weight on the average". Finally do we include your definition stated here.