A fractional coloring of $G(V,E)$ is a function $f:{2^{|V|}} \to [0,1]$ such that $f(S) > 0$ only if $S$ is an independent set in $G$, and for all $v \in V$, it holds $\sum\nolimits_{S:v \in S} {f(S) \ge 1} $. A fractional chromatic number $\chi_f(G)$ is the minimum $k$ for which there exists a fractional coloring $f$ for $G$ with $\sum\nolimits_S {f(S) = k} $.
Let $G = (V,E) $ be a graph and let $k \in \mathbb{N}$. Show that ${\chi_f}(G) \le k$ if and only if for all weight function $w:V \to \mathbb{R}_+$ there is an independent set of size at least $\left( {\sum\nolimits_{v \in V} {w(v)} } \right)/k$. Moreover, if we are given an algorithm that computes an independent set of size $w(V)/k$ for all weight function $w$, then this algorithm can be used to compute a fractional coloring using $(1 + \varepsilon )k$ colors.
Where i can read about the proof of this claim?
