This question is related to the one in https://mathoverflow.net/questions/65322/the-structure-of-certain-maximal-sets-of-means-into-amenable-semigroups. I open a different topic because they can be treated separately.

Let $S$ be a countable amenable semigroup and let $IM(S)$ be the set of the invariant means on $S$.

Definition.For any subset $W\subseteq S$, we define $$ W^-=\inf_{m\in IM(S)}m(\chi_W)\;\;\;\;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;\;\;\;\;\;W^+=\sup_{m\in IM(S)}m(\chi_W) $$

There are many and well-known examples that show that these numbers could be different. I am interested in the case when they are equal. As far as I know it seems to me that this situation has no name in literature, so let me fix the following terminology

Definition. $W$ is said to have the propertyIM(Intrinsically Measurable) if $W^-=W^+$. In this case $\mu(W)$ denotes theintrinsic measureof $W$.

Basic examples of sets with IM are easy. I need to give such examples in order to formulate my first question. First of all, recall the following (almost) classical

DefinitionLet $k$ be a positive integer possibly infinite. A subset $W\subseteq S$ is called $k$-tile if there are $s_1, s_2,... s_k$ elements in $S$ such that

- $s_iW\cap s_jW=\emptyset$, for all $i\neq j$
- $S\setminus\bigcup s_iW$ has the property IM and $\mu(S\setminus\bigcup s_iW)=0$

Tiles obviously have the property IM. Moreover the following operations preserve the property IM: finite union of disjoint sets with IM; if $V\subseteq T$ have the property IM, then also $T\setminus V$ have IM. Let me call **elementary** those sets with the IM that can be obtained by tiles using the previous operations.

Now I list some basic questions about this property IM. The second question is not just a specification of the first one, but it is interesting for the application that is the motivation for studying this property.

Question 1.Is there an (explicit) example of a non-elementary IM set $W$ with $\mu(W)>0$?

Question 2.Let $S$ be the multiplicative semigroup of positive integers and let $W$ be the subset of those positive integers with first digit $1$, $2$ or $3$. Does $W$ have the property IM?

Note that the property IM is not closed under countable union, but even the answer to the following question is not completely evident to me

Question 3.Is the class of IM sets closed under finite intersection? [This question has been answered in the negative by Ben Willson (see below)]

Thanks in advance for any comment,

Valerio