Here is a simpler argument, combining 1--6 into one step.
Let $G$ be a countable abelian group generated by $x_1,x_2,\ldots$. Then a Følner sequence is given by taking $S_n$ to be the pyramid consisting of elements which can be written as
$a_1x_2+a_2x_2+\cdots+a_nx_n$ with $\lvert a_1\rvert\leq n,\lvert a_2\rvert\leq n-1,\ldots,\lvert a_n\rvert\leq 1$.
The invariant probability measure is then defined by $\mu(A)=\underset{\omega}{\lim}\lvert A\cap S_n\rvert / \lvert S_n\rvert$ as usual.
A more natural way to phrase this argument is:
- The countable group $\mathbb{Z}^\infty$ is amenable.
- All countable abelian groups are amenable, because amenability descends to quotients.
But I would like to emphasize that there is really only one step here, because the proof for $\mathbb{Z}^\infty$ automatically applies to any countable abelian group. This two-step approach is easier to remember, though. (The ideas here are the same as in my other answer, but I think this formulation is much cleaner.)
2016 Edit: Here is an argument to see that $S_n$ is a Følner sequence. It is quite pleasant to think about precisely where commutativity comes into play.
Fix $g\in G$ and any finite subset $S\subset G$. We first analyze the size of the symmetric difference $gS\bigtriangleup S$. Consider the equivalence relation on $S$ generated by the relation $x\sim y$ if $y=x+g$ (which is itself neither symmetric, reflexive, or transitive). We will call an equivalence class under this relation a "$g$-string". Every $g$-string consists of elements $x_1,\ldots,x_k\in S$ with $x_{j+1}=x_j+g$.
The first key observation is that $\lvert gS\bigtriangleup S\rvert$ is at most twice the number of $g$-strings. Indeed, if $z\in S$ belongs to $gS\bigtriangleup S$, then $z$ must be the "leftmost endpoint" of a $g$-string; if $z\notin S$ belongs to $gS\bigtriangleup S$, then $z-g$ must be the "rightmost endpoint" of a $g$-string; and each $g$-string has at most 2 such endpoints (it could have 1 if the endpoints coincide, or 0 if $g$ has finite order).
Our goal is to prove for all $g\in G$ that $\frac{\lvert gS_n \bigtriangleup S_n\rvert}{\lvert S_n\rvert}\to 0$ as $n\to \infty$. Since $\lvert abS\bigtriangleup S\rvert\leq\lvert abS\bigtriangleup bS\rvert+\lvert bS\bigtriangleup S\rvert= \lvert aS\bigtriangleup S\rvert+\lvert bS\bigtriangleup S\rvert$, it suffices to prove this for all $g_i$ in a generating set.
By the observation above, to prove that $\frac{\lvert g_iS_n \bigtriangleup S_n\rvert}{\lvert S_n\rvert}\to 0$, it suffices to prove that $\frac{\text{# of $g_i$-strings in $S_n$}}{\lvert S_n\rvert}\to 0$. Equivalently, we must prove that the reciprocal $\frac{\lvert S_n\rvert}{\#\text{ of $g_i$-strings in $S_n$}}$ diverges, or in other words that the average size of a $g_i$-string in $S_n$ diverges.
We now use the specific form of our sets $S_n=\{a_1g_1+\cdots+a_ng_n\,|\, \lvert a_i\rvert\leq n-i\}$. For any $i$ and any $n$, set $k=n-i$ (so that $\lvert a_i\rvert\leq k$ in $S_n$). The second key observation is that every $g_i$-string in $S_n$ has cardinality at least $2k+1$ unless $g_i$ has finite order. Indeed given $x\in S_n$, write it as $x=a_1g_1+\cdots+a_ig_i+\cdots+a_ng_n$; then the elements $a_1g_1+\cdots+bg_i+\cdots+a_ng_n\in S_n$ for $b=-k,\ldots,-1,0,1,\ldots,k$ belong to a single $g_i$-string containing $x$. If $g_i$ does not have finite order, these $2k+1$ elements must be distinct. This shows that the minimum size of a $g_i$-string in $S_n$ is $2n-2i+1$, so for fixed $g_i$ the average size diverges as $n\to \infty$.
When $g_i$ has finite order $N$ this argument does not work (a $g_i$-string has maximum size $N$, so the average size cannot diverge). However once $N<2k+1$, the subset containing the $2k+1$ elements above is closed under multiplication by $g_i$. In other words, once $n\geq i+N/2$ the set $S_n$ is $g_i$-invariant, so $\lvert g_iS_n\bigtriangleup S_n\rvert=0$.
I'm grateful to David Ullrich for pointing out that this claim is not obvious, since the quotient of a Følner sequence need not be a Følner sequence (Yves Cornulier gives an example here).
Your proof can be modified a bit so that it works for a general countable, amenable group $\Gamma$. In this case, we can take $B(X)$ to be the closure of $\mbox{span} \{ f \circ T_{\gamma} - f : f \in C(X), \gamma \in \Gamma \}$ and we show that $B(X)$ doesn't contain the constant functions on $X$ as follows:
I claim that every function $h \in \mbox{span} \{ f \circ T_{\gamma} - f : f \in C(X), \gamma \in \Gamma \}$ is non-negative at some point in $X$, namely $\max_{x \in X} h(x) \ge 0$. This is sometimes called the "translate property" for amenable groups, if I'm not mistaken. Anyway, after establishing that, the conclusion follows easily: it follows that one cannot uniformly approximate a negative constant (say $-7$) by a function of the above form.
The proof of the translate property I'm familiar with goes like this:
Assume for contradiction that $\max_{x \in X} h(x) = \delta < 0$ and let $\varepsilon > 0$. Let $h=\sum_{i=1}^{n}f_{i}\circ T_{\gamma_{i}}-f_{i}$. Since $\Gamma$ is amenable we can find a finite set $U \subset \Gamma$ such that $\frac{\left|\gamma_{i}U\triangle U\right|}{\left|U\right|}<\varepsilon$ for all $i=1,\dots,n$ (the Følner condition). Now consider the function $F:=\frac{1}{\left|U\right|}\sum_{u\in U}h\circ T_{u}$. Choose some point $x_0 \in X$. On the one hand, by our assumption, $F\left(x_{0}\right) = \frac{1}{\left|U\right|}\sum_{u\in U} h \left(T_{u}x_{0}\right)\le\delta<0$. On the other hand,
$\left|F\left(x_{0}\right)\right|=\left|\frac{1}{\left|U\right|}\sum_{u\in U}\sum_{i=1}^{n}f_{i}\left(T_{\gamma_{i}}T_{u}x_{0}\right)-f_{i}\left(T_{u}x_{0}\right)\right|$
By the triangle inequality, changing order of summation and using the property of U, you see that the above is at most $2n \varepsilon \max_{x\in X,1\le i\le n}\left|f_{i}\left(x\right)\right|$. So if you choose $\varepsilon$ small enough, you get a contradiction.
Best Answer
Here is a sketch of an argument to show these properties are equivalent which is based on the proof of the same property for amenable groups. To see this I will use the fact that an action $G \curvearrowright (X, \mu)$ is amenable if and only if there is a $G$-equivariant mean from $L^\infty(X, \mu) \overline \otimes L^\infty(G)$ to $L^\infty(X, \mu)$. (A mean in this setting is a positivity preserving map which restricts to the identity on $L^\infty(X, \mu)$.) This is Theorem A (iv) in "Amenable Actions of Groups" by Adams, Elliott, Giordano (1994) (http://www.ams.org/mathscinet-getitem?mr=1250814), which generalizes the case for countable groups established by Proposition 4.1 in "Hyperfinite Factors and Amenable Ergodic Actions" by Zimmer (1977) (http://www.ams.org/mathscinet-getitem?mr=470692).
Theorem: Let $G$ be a locally compact second countable group. Then there exists a $G$-equivariant mean from $L^\infty(X, \mu) \overline \otimes L^\infty(G)$ to $L^\infty(X, \mu)$ if and only if for every continuous action of $G$ on a compact metric space $Y$, there exists a $G$-equivariant map from $X$ to $M(Y)$.
Proof: For the proof I will find it easier to work with function spaces, and so first note that for a compact Hausdorff space $K$, there is a one-to-one correspondence between (equivalence classes of) Borel maps $\pi: X \to M(K)$ and means $\Phi: L^\infty(X, \mu) \otimes C(K) \to L^\infty(X, \mu)$. (Here, $\otimes$ is the $C^*$-tensor product.) This is essentially just the Riesz representation theorem applied to each fiber $K$.
We now show the "only if" direction. If $G \curvearrowright K$ is a continuous action on a compact metric space, by restricting to a closed $G$-invariant subset we may assume that $G \curvearrowright K$ has a dense orbit $K = \overline{G k}$. We then obtain a $G$-equivariant embedding $\rho: C(K) \to C_b(G) \subset L^\infty(G)$ by $\rho(f)(g) = f(gk)$. Restricting our invariant mean to $L^\infty(X, \mu) \otimes \rho(C(K))$ then gives the result.
The converse is a bit more involved, but this is well known for the case of amenable groups. The first step is to show that we have an equivariant mean on the space $UC_b(G)$ of bounded left uniformly continuous functions. In the case when $G$ is countable we are then done. For the general case we then use an approximate identity for convolution to produce an equivariant mean for $L^\infty(G)$.
To produce a mean on the space $UC_b(G)$ note that for any second countable $G$-invariant $C^*$-subalgebra $A \subset UC_b(G)$ we have that the Gelfand spectrum is metrizable, and $G$ acts continuously. Therefore by hypothesis there exists a $G$-equivariant mean $\Phi_A: L^\infty(X, \mu) \otimes A \to L^\infty(X, \mu)$. Since $L^\infty(X, \mu)$ is injective we may then extend $\Phi_A$ to a (perhaps no longer $G$-equivariant) positivity preserving map $\widetilde{\Phi_A} : L^\infty(X, \mu) \otimes UC_b(G) \to L^\infty(X, \mu)$. If we consider the net $\{ \widetilde{\Phi_A} \}$ indexed by the set of all second countable $G$-invariant $C^*$-subalgebras $A$, and ordered by inclusion, then we may take an accumulation point $\Phi$ in the topology of point-wise weak convergence. Since $\Phi$ is $G$-equivariant when restricted to $L^\infty(X, \mu) \otimes A$ for any second countable $C^*$-subalgebra $A$, it follows that $\Phi$ is $G$-equivariant on the whole space.
To produce a mean on $L^\infty(X, \mu) \overline \otimes L^\infty(G)$ we start by taking an approximate identity $\{ \psi_n \} \subset C_c(G)$. Specifically, we want that each $\psi_n \in C_c(G)$ is a non-negative function, $\| \psi_n \|_1 = 1$, ${\rm supp}(\psi_n) \to \{ e \}$, and $\| \psi_n * \delta_g - \delta_g * \psi_n \|_1 \to 0$ for each $g \in G$. If $f \in L^\infty(X, \mu) \overline \otimes L^\infty(G)$, then we have $\psi_n * f \in L^\infty(X, \mu) \otimes UC_b(G)$ for each $n$ (I'm taking convolution pointwise in $X$), and $\| \sigma_g(\psi_n * f) - \psi_n * (\sigma_g(f)) \|_\infty \to 0$ for each $g \in G$, and $f \in L^\infty(X, \mu) \overline \otimes L^\infty(G)$. (We denote by $\sigma_g$ the action of $G$ on $L^\infty(X, \mu) \overline \otimes L^\infty(G)$.) If we set $\Phi_n : L^\infty(X, \mu) \overline \otimes L^\infty(G) \to L^\infty(X, \mu)$, by $\Phi_n( f ) = \Phi( \psi_n * f )$, then it follows that any accumulation point of $\{ \Phi_n \}$ gives us a $G$-equivariant mean. $\blacksquare$
Update:Nicolas Monod pointed out to me that, in my sketch above, the way that I use the approximate identity $\{ \psi_n \}$ at the end is not correct. So the proof that I outline seems to work just fine for discrete groups, but the general case for locally compact groups appears more difficult. In fact, it appears to be related to the notion of "relative amenability" presented in:
Caprace, Pierre-Emmanuel; Monod, Nicolas: Relative amenability. Groups Geom. Dyn. 8 (2014), no. 3, 747–774.
Update 2: Even with discrete groups this answer only works with the $C^*$-tensor product. However, the answer by Buss points to their paper containing a very satisfying resolution of this issue through the use of exactness.