I am confused by the definition of Lovasz extension. The problem is I don't get the intuition behind the definition. In addition, Lovasz extension can be defined in different ways I don't see that these definition are indeed equivalent. The following is the definition and few my question with the current level of understanding.
For a function $f:\{0,1\}^N \rightarrow R, f^L : [0,1]^N \rightarrow R$ is defined by
$f^L(x) = \sum_{i=0}^{n} \lambda_if(S_i)$
where $\varnothing =S_0 \subset S_1 \subset S_2 \subset ... \subset S_n$ is a chain such that $\sum_{}^{} \lambda_i 1_{S_i}=x$ and $\sum_{}^{} \lambda_i=1, \lambda_i \geq 0$
An equivalent way to define the Lovasz extension is :$f^L(x) = E[f(\{i:x_i > \lambda\})]$, where $\lambda$ uniformly random in $[0,1]$
Question 1. Intuition. I don't get the intuition. $f$ is some valuation of function defined on all subsets of $N$ elements. $f^L(x)$ is the similar for $f$ but as input $x$ can take fractional values of elements.
Question 2: What the following function means $\sum_{}^{} \lambda_i 1_{S_i}=x$? In particular I don't get what's $1_{S_i}$ means. It looks like a linear combination of $\lambda_i$ and a magic stuff $1_{S_i}$ (please, what does it mean?) such that as result we get the input which is the vector of $N$ elements with fractional values.
Question 3: Equivalent definition is $f^L(x) = E[f(\{i:x_i > \lambda\})]$, it's expected value, does it mean I can rewrite it as follows $f^L(x) = x_if(i), \forall x_i > \lambda$, it this the same $\lambda$ as in the first definition.
Question 4: the requirement $\varnothing =S_0 \subset S_1 \subset S_2 \subset ... \subset S_n$ seems rather strange, what the goal of this requirement.