Do lowpass filters affect the integral over the signal? For which filters, does the resulting signal have the same integral (/average) as the original? Obviously this is true for a moving average, but is it also mathematically true for e.g. a Butterworth filter?

I am trying to build a lowpass filter for a signal of measurement samples. While there is a lot of noise, the average (over a sizable number of samples) is correct. I need to filter it retaining that property, regardless of the input frequency spectrum.
You need a filter whose transfer function evaluated at 0 (DC component) is 1, or equivalently, whose impulse response sums to 1.