A question about measuring white noise in practice
When you come to measure the wideband noise of a system, you start with a definition of the bandwidth you're interested in.

How do you define that bandwidth?

It may be a bandwidth that's relevant to your use (say 20Hz to 20kHz if it's an audio application), or it may be that it's limited by your test gear (my 'scope only goes up to 50MHz), but somehow, somewhere, you have to choose a bandwidth. If you're building your own test gear, or you don't know enough yet about your system or application, then it may be an educated guess.

Once you've chosen your bandwidth, you put an anti-alias filter in front of your digitiser (or maybe it's there already, defining your bandwidth). The AA filter passes the whole band of interest, and protects the digitiser from aliasing.

If you don't realise that the measuring instrument has chosen a bandwidth for you, then you may run into calculation or interpretation problems later.

What do you do next? 

If your measurements of noise correlate with the behaviour of your system, so it allows you to predict behaviour, design improvements, reject and investigate faults, then it's job done. If you find it doesn't, or suspect there's noise at a higher frequency than you're measuring, then you have to revise your guess. Build, buy or rent better test gear, and measure it to a higher frequency. Rinse and repeat.

This iterative method of seeing what bandwidth is relevant is useful, as it means you don't have to measure to infinite frequency for every job. Start with the simplest measurement that you hope will do the job, then increase cost and complexity if you find it's necessary.