Non-stop ADC for full quantization of arbitrary time period signal average. Do ADCs work like this?
Most types of analog to digital converters have intervals of time when they are sampling their input, and intervals of time when they are not.  If one needs a time-averaged signal, there are two ways of accomplishing this:


Filter the signal significantly before sampling; if the cutoff frequency for the filter is well below half the sampling rate, the output of the filter will not vary much in the interval between samples.  Consequently, no matter when samples are taken they will be relatively close to the average signal value.
A Delta-Sigma modulator, as MarkU noted, samples continuously.  Not all delta-sigma ADCs do so, however.  If a delta-sigma ADC counts off fixed intervals and generates a reading each time, then it may be able to guarantee that a long-term average value of the readings will equal the average value of the input signal.  If the converter only generates readings when triggered, however, or if not all of its readings are captured, then such a guarantee would not hold.


Both filtering and the use of delta-sigma converters may offer useful averaging behaviors.  Which is better in a particular application will depend upon one's exact requirements.