It is assumed that in one of the lectures,
$$\int_\mathbb{R} e^{-iux}\,I(x=0)\,dx = 1$$
Shouldn't this formula be equal to $\mathbf{dx}$ instead of $\mathbf{1}$? This formula has non-zero value only when $x=0$, so $$e^{-iu0}\,1\,dx = \mathbf{dx}$$ will be the final result of that integration instead of $\mathbf{1}$. If I am missing any concept, please let me know.