# Why use indicator variables in p.d.f.s?

I am slightly confused about the use of indicator variables in probability density functions.

For example, consider the density of $$(X,Y)$$ uniformly distributed on the unit disc. This density can be written as $$f_{X,Y}(x,y)= \frac{1}{\pi} \mathbb{1} \left \{ (x,y) \mid x^2 + y^2 \leq 1 \right \}.$$

However, I am not really seeing how this makes sense. I understand indicator variables when solving problems with probability, but why not just write the density without it? What is it exactly saying?

Note that $$f_{X,Y}:\mathbb R^2\to\mathbb R$$.
It says exactly that $$f_{X,Y}(x,y)$$ will take value $$\frac1{\pi}$$ if $$x^2+y^2\leq1$$ and will take value $$0$$ otherwise.
So an excellent presentation of a PDF which is formally for a fixed nonnegative integer $$n$$ a nonnegative measurable function $$\mathbb R^n\to\mathbb R$$ that gives value $$1$$ by integration wrt Lebesgue-measure.
In notation it can be handsome to write things like: $$\mathbb EX=\int f_X(x)xdx$$ without bothering on borders.
If it is not your taste then of course you can also choose for discerning cases: $$x^2+y^2\leq1$$ and "otherwise" without any mentioning of indicatorfunction.