The natural generalization of Haar measure onto semigroups doesn't feel harmonious (as even the material already provided in this thread seems to indicate). Thus let me introduce two other notions: *strict* and *liberal*. When a semigroup is actually a group then its liberal measures (henceforth the strict too) are invariant. By *invariance* I mean a one-sided invariance (I don't know which side though since I am ambidextrous :-)
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**NOTATION** Let $\ (G\ \cdot)\ $ be a semigroup. Let $\ c\in G.\ $ Then let $\ m_c : G\rightarrow G\ $ be defined by: $\ m_c(x) = x\cdot c\ $ for every $\ x\in G$.
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Let $\ (G\ \cdot)\ $ be a semigroup. Let $\ \mu\ $ be a $\sigma$-measure in $G$ (with its respective $\sigma$-algebra). Measure $\mu$ is called *liberal* $\ \Leftarrow:\Rightarrow\ $ for every measurable set $\ A\subseteq G\ $ such that $\ m_c^{-1}(m_c(A))\ = A\ $, and for every $\ c\in G,\ $ there exists a measurable $\ B\subseteq G\ $ such that $\ m_c(A)\subseteq B\ $ and $\ \mu(A)\ge\mu(B)$.

Also, measure $\ \mu\ $ is called *strict* $\ \Leftarrow:\Rightarrow\ $ for every measurable $\ A\subseteq G\ $ there exists a measurable $\ B\subseteq G\ $ such that $m_c(A)\subseteq B\ $ and $\ \mu(A)\ge \mu(B)$.
 
Thus every strict measure is liberal is strict (while there are simple examples of non-strict liberal measures). Nevertheless, even in the liberal case, they are all invariant in the case of any group. Next, in the case of finite semigroups, cardinality is a strict measure. It'd be awesome (:-) to connect the general liberal and strict measures with the structure of finite semigroups. And in the case of (infinite) topological semigroups we would get a lot to think about it seems.