Given a graph $G=(V,E)$. A fractional matching, say $f$, assigns every edge $e \in E$ to a fraction $f(e) \in [0,1]$, with the constraint: for $v \in V$, $\sum_{e \ni v}f(e) \leq 1 $.
My question is : does the fractional matching give a matching that we know (of $G$) i.e. with no conflicting edges? I am confused, since there is no criteria for choosing a fraction $f(e) \in [0,1]$, for every $e \in E$. For example, if we have $e_{1}, e_{2} \in E$ such that $v \in e_{1}, e_{2} $ and I assign $f(e_{1})=1/4$ and $f(e_{2})= 1/2$, then $f(e_{1})+f(e_{2})=3/4 \leq 1 $, hence both $e_{1}$ and $e_{2}$ should belong to the fractional matching.
Thank you in advance.