Confused by the notation in $\Omega(g) =\infty\mathbb{1}[\|w_g\|_0>k]$

Question

$$\Omega(g) =\infty\mathbb{1}[\|w_g\|_0>k]$$

(original image)

(Note: The "$\Bbb{1}$" should be in the blackboard-bold style of, say, $\Bbb{Z}$, but it doesn't seem to render that way for me. --Blue)

I am sorry that I am confused by the symbol of Double-Struck Digit One, infinity symbol and the zero below the $w_g.$ Does one means identity matrix here or the function?:

• $1$ belong to set
• $0$ not belong to set

Why there is infinity before one?

What is the meaning of that zero outside the Normal Form?

This is from the paper "Why Should I Trust You?" Explaining the Predictions of Any Classifier (PDF link via arxiv.org); p4, left, upper part

Answer 1

• $\mathbb{1}$ is the notation for the indicator function.

• The $\infty$ is multiplied to the indicator function.

• $\|v\|_0$ counts how many non-zero entries are there in $v$.