But how do we define mean(f) in this case?
If the data is discrete, looks like there's no need to find the integration -- we just add the discrete parts together.
But in the continuous case, it feels like we need the integral to find mean(f), which makes the argument "circular" here.

But how do we define mean(f) in this case? If the data is discrete, looks like there's no need to find the integration -- we just add the discrete parts together. But in the continuous case, it feels like we need the integral to find mean(f), which makes the argument "circular" here.