Haar Measure and the Semigroup of Measures on a Compact Group, J. G. Wendel,
Proceedings of the American Mathematical Society
Vol. 5, No. 6 (Dec., 1954), pp. 923-929
proves the existence of Haar measure in compact groups using the existence of idempotents and minimal ideals in compact semigroups.
In more detail, the probability measures on a compact group form a compact semigroup with respect to convolution of measures and the weak* topology. One easily checks that the support of an idempotent measure is a closed subgroup and that the measure is a Haar measure on the support. If one takes an idempotent measure in the minimal ideal one checks the support is global and hence the measure is Haar measure for the group. Since all compact semigroups have a minimal ideal and the minimal ideal must have an idempotent, this yields Haar measure exists for compact groups.