Sorry, my intent was to understand what the equivalent circuit is at the plug. Can the grid simply be modeled as an AC source between "L" and "N", and "N" is connected to earth?
If the grounds are independent then it seems there would be an issue taking measurements as a scope ground would be the same as the grid ground.
Yes!
This is something that comes up in this forum a lot.
If you have to connect between two points in such a circuit where neither are at the mains protective earth potential,you need to use either an Oscilloscope with inputs which are not referenced to mains earth,two channels in a "quasi differential connection" or use a differential probe.
If you are basically just "troubleshooting" in a circuit like this,you can remove your ground clip from the probe entirely,& just probe the various points,using the 'scope's earthing via its power cord as the return path.
Obviously,if you need to see high frequency components,this is not satisfactory,as you have tens of metres of wire in the round trip via the Neutral/Earth connection at the power entry point.