= Solution
For a discrete <group>, one standard definition of amenability is the fixed-point property: every continuous affine action on a nonempty compact <convex set> in a locally convex space has a fixed point. The equivalent state formulation, which may be stated without proof here, is that there is a left-invariant <invariant mean> on $\ell^\infty(\Gamma)$. Equivalently, every <unital> left-translation-invariant <star-subalgebra> of $\ell^\infty(\Gamma)$ admits a left-invariant state, with <continuity> for the supremum <norm>; one may equally pass to its <norm> <closure>. A state means a positive <linear functional> of <norm> one with value one on the constant function one. Restriction proves the <subalgebra> formulation from the full-algebra version, and taking the full algebra proves the converse. Such a mean is finitely additive on <indicator functions>, not necessarily countably additive.
Write $(\lambda(g)\mu)(h)=\mu(g^{-1}h)$ on $\ell^1(\Gamma)$ and $(L_gf)(h)=f(g^{-1}h)$ on bounded functions. We prove equivalence of this mean condition, the <Reiter condition>, and existence of a <Følner sequence>.
Assume first that an <invariant mean> $m$ exists. Fix a finite set $K\subseteq\Gamma$. In the real <Banach space> $\bigoplus_{g\in K}\ell^1(\Gamma)$ with sum <norm>, consider the <convex set>
$$
\mathcal C=\{(\lambda(g)\mu-\mu)_{g\in K}:\mu\text{ a finitely supported probability measure}\}.
$$
If zero were outside its <norm> <closure>, the <Hahn-Banach separation theorem> would give bounded real functions $f_g$ and $\delta>0$ such that
$$
\sum_{g\in K}\sum_h f_g(h)[(\lambda(g)\mu)(h)-\mu(h)]\geq\delta
$$
for every such $\mu$. Testing point masses at $h$ shows
$$
\sum_{g\in K}[f_g(gh)-f_g(h)]\geq\delta\quad\text{for every }h.
$$
Applying the positive <invariant mean> makes the left side zero and the right side at least $\delta$, a contradiction. Therefore zero is in the <closure>: for any $\varepsilon>0$ a finitely supported <probability measure> makes the sum of the translation defects over $K$ less than $\varepsilon$. Enumerate the countable <group>, take the first $n$ elements for $K$ and $\varepsilon=1/n$, and obtain a sequence satisfying the <Reiter condition>. This proves (1) implies (2).
Next assume (2). An $\ell^1$ <probability measure> can be approximated in <norm> by finitely supported <probability measures>, by restricting to a large finite set and renormalizing. Replacing $\mu$ by $\nu$ changes each defect by at most $2\|\mu-\nu\|_1$. Thus simultaneous small defects for finitely many translations can be achieved with finite support.
For such $\nu$, set $F_t=\{h:\nu(h)>t\}$. The elementary identity $|r-s|=\int_0^\infty|1_{r>t}-1_{s>t}|\,dt$ for $r,s\geq0$ gives
$$
\int_0^\infty|F_t|\,dt=1,\qquad
\int_0^\infty|F_t\mathbin\triangle gF_t|\,dt
=\|\lambda(g)\nu-\nu\|_1.
$$
If the sum of defects over $K$ is less than $\varepsilon$, some $t$ with $F_t\ne\varnothing$ must satisfy $\sum_{g\in K}|F_t\triangle gF_t|<\varepsilon|F_t|$; otherwise integration would contradict that strict inequality. Applying this <layer-cake extraction of Følner sets> to successive finite sets of translations gives nonempty finite $F_n$ with each relative boundary tending to zero. Hence (2) implies (3).
Conversely, normalized counting measures $\mu_n=1_{F_n}/|F_n|$ satisfy
$$
\|\lambda(g)\mu_n-\mu_n\|_1=\frac{|gF_n\triangle F_n|}{|F_n|},
$$
so (3) implies (2). Finally, from (2) define states $m_n(f)=\sum_h\mu_n(h)f(h)$. The dual <unit ball> of $\ell^\infty(\Gamma)$ is weak-star compact by the <Banach-Alaoglu theorem>; take a convergent subnet, since this <compactness> need not be metrizable. Its limit $m$ is positive with $m(1)=1$, and
$$
|m_n(L_gf)-m_n(f)|\leq\|f\|_\infty\|\lambda(g^{-1})\mu_n-\mu_n\|_1\longrightarrow0.
$$
Thus $m$ is invariant, proving (2) implies (1). Together these implications establish all three equivalences. Countability is used to obtain sequences from finite tests; it is not a justification for replacing the <compactness> subnet by a subsequence.
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