In this paper we will characterize the left $X$-thick subsets of an $X$-amenable topological semigroup $S$. Our result extends a result of $T$. Mitchell in $[12]$ and Wilde and Witz $[15]$.

simultaneous generalization of the normal Radon measures of Dixmier and the category measures of Oxtoby. We examine the regularity, $\tau$-smoothness, tightness, and support properties of residual measures. We show that residual measures without support exist iff real-valued measurable cardinals exist. In the compact setting we associate with any compact Hausdorff space $X$ a larger Stonian compact Hausdorf space, the Gleason space of $X$, such that there is a bijective correspondence between the residual measures on these spaces and the residual Radon measures on these spaces. Hence, we lift the question of existence of certain types of residual measures to the Stonian setting of Dixmier.

The Rudin-Shapiro coefficients $\{a(n)\}$ are an infinite sequence of $\pm 1$'s, defined recursively by $a(0)=1$, $a(2n)=a(n)$, and $a(2n + 1)=(-1)^{n}a(n)$, $n \geq 0$ Various formulas are developed for the $n$th partial sum $s(n)$ and the $n$th alternating partial sum $t(n)$ of this sequence. These formulas are then used to show that $\sqrt{3/5} < s(n)/\surd n < \surd 6$ and $0 \leq; t(n)/\surd n < \surd 3$, $n \geq 1$ where the inequalities are sharp and the ratios are dense in the two intervals. For a given $n \geq 1$, the equation $s(k)= n$ is shown to have exactly $n$ solutions $k$.