probability - For all non-negative random variables, why is $X=int_0^{+infty}mathbf 1_{Xgeq t},mathrm dt$ true?

I am wondering why this equation is necessarily true for all non-negative random variables:

$$X=\int_0^{+\infty}\mathbf 1_{X\geq t}\,\mathrm dt$$

What is confusing me is that It appears that the indicator function only spits out a value of $1$ and that I am not seeing the connection here and how the integral over the indicator function makes it $X$. Thanks!

Answer

I'm not seeing why you would be confused unless you are trying to integrate with respect to $X$ instead of $t$.

$$\int\limits_{0}^\infty\mathbf 1_{X>t}\operatorname d t~=~\int\limits_{0}^\infty\mathbf 1_{t

Which is of course $X$ when $X>0$