For Haar measure and other results, see Valenza and Ramakrishnan: Fourier analysis on Number Fields
Serre: Bourbaki exp 27. Nice results of Iwasawa and Gleason
Borel: Bourbaki exp 29.
LNM0002 treats cohomology of locally compact spaces. The treatment looks “elementary” and pre-Grothendieck.
nLab page on Locally compact groups