This thesis focuses on the measure algebra M(S) of a locally compact semitopological semigroup S. In particular, we consider the analog of the group algebra L1(G) of a locally compact group G on S and the topological amenability of S. Among other results which shall be explained further in the introduction, the thesis answers the following open problems. 1. Baker 90' and Dzinotyiweyi 84' [6, 18] Let L(S) = {μ ∈ M(S); s →δs*|μ| is weakly continuous}. It is known that if S = G, then L(S) = L1(G). Is L(S) a norm closed ideal of M(S) that closed under absolute continuity in general? We shall answer this question in the positive in Section 3.3 and 3.4. 2. Day 82' [15] We say S is strong topological left amenable if there is a net of probability measure (μα) such that

→ 0 uniformly for all probability measures _ supported on a compact subset K of S. Does strong topological amenability implies non-trivial L(S)? The background for this question will be explained fully in Section 4.1, along with a counterexample that answers this problem in the negative. 3. Wong 79' [50] We say S is topological left amenable if there is a net of probability measure (μα) such that

→ 0 for any probability measure ν on S. It was shown that when S is a discrete semigroup or a locally compact group, a locally compact Borel subsemigroup T is topological left amenable if and only if (1) S is topological left T-amenable, that is, there is a net of probability measures (μα) on S, such that

→ 0 for any probability measure _ on S that is supported on T, and (2) limα μα (T) > 0. Does similar result hold for locally compact semitopological semigroups? We shall prove this result in Section 4.2.

ν* μα- μα

ν* μα- μα

ν* μα- μα