Why is $1_{\{\omega : X(\omega) \geqslant t\}}(\omega) = 1_{\{t: t\leqslant X(\omega)\}}(t)$.

Question

I cannot see the logic behind how $1_{\{\omega : X(\omega) \geqslant t\}}(\omega) = 1_{\{t: t\leqslant X(\omega)\}}(t)$ in the proof here, Expectation of $\mathbb{E}(X^{k+1})$

Could someone please explain this in more detail.

Answer 1

Both $1_{\{\omega : X(\omega) \geqslant t\}}(\omega)$ and $1_{\{t: t\leqslant X(\omega)\}}(t)$ are $1$ when $X(\omega) \geqslant t$ and $0$ otherwise. The only difference between the two expressions is that on the left you think of this as a function of $\omega$ and on the right you think of it as a function of $t$.