Two open problems of Day and Wong on left thick subsets and left amenability

This paper answers two open problems raised by Mahlon M. Day [4] and James C.S. Wong [17]: Does uniformly topological left amenability imply the existence of left translation continuous measures on a locally compact semitopological semigroup? Let T be a locally compact Borel subsemigroup of a locally compact semitopological semigroup S. Is the existence of a topological T-invariant mean M on S with M(χT)>0 enough to imply the topological left amenability of T? We give a negative answer to the first question by providing a counterexample and a positive proof to the second. This example can also be used to show that the property (α) provided in Gerard L. Sleijpen [14] can not be removed in one of his results.