I am currently working through notes on Sobolev Spaces in Partial Differential Equations by Lawrence.c.Evans

Check 2 page attachment for relevant notes. It is the proof of the Trace Theorem.

In the proof I would like to know why (1) is not trivial since and is a boundary which therefore has measure zero, does it not follow then that , therefore .

And then would the last line not also follow easily since .

Thanks