I am reading the wikipedia article about submodular functions. Let $\Omega$ be a finite set and $f\colon 2^\Omega\to \Bbb R$ a submodular set function, i.e. a function such that $$f(S)+f(T)\geq f(S\cup T)+f(S\cap T) \qquad \forall S,T\subset \Omega$$ Note that there are other equivalent characterizations in the article. Now, the article talks about a multilinear extension $F\colon [0,1]^{|\Omega|}\to \Bbb R$ of $f$ (here) defined by $$F(x)=\sum_{S\subset \Omega}f(S)\prod_{i\in S}x_i\prod_{i\notin S}(1-x_i).$$
How is $F$ multilinear and how to check it?
I tried to build an example for $\Omega = \{1,2,3\}$ and the budget additive function $f(S)=\min\{3,\sum_{\tilde s\in S}\tilde s\}$ (e.g. $f(\{1,2\})=\min\{3,1+2\}$). I get $F(x,y,z)=x+2y+3z-xz-2yz$ and don't see any way how could $F$ be multilinear.