I was looking at
Izzo, Alexander J., A functional analysis proof of the existence of Haar measure on locally compact Abelian groups, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). ZBL0777.28006.
which proves existence of the Haar-measure for locally compact abelian groups using the Markov-Kakutani theorem.
What I find strange is that the Haar measure is constructed as an element of the dual of $C_c(X)$. But for noncompact $X$ (such as $X$ being the real numbers $\Bbb R$) this must be an unbounded functional (as the Lebesgue-measure on $\Bbb R$ is not finite). It seems like the author has no problem with this, and (without mentioning it further) goes on to define a weak-* topology for this case and even uses Banach-Alaoglu.
I have not seen this being done this way before, am I misunderstanding something or can one define a weak-* topology on the algebraic dual of a TVS without any problems?