We may approximate a bounded nonnegative function $f$ by simple functions $\phi$. Then the standard way of defining the Lebesgue integral of $f$ is$$ \int f d\mu := \sup \Bigg\{ \int \phi : \phi \leq f, \phi \textrm{ is simple}\Bigg\}.$$I am very comfortable with this definition but it was not until recently while learning about Ito integrals that I began thinking about why the Lebesgue integral is not defined as a limit.
Loosely speaking, if $X_t$ is a stochastic process and $B_t$ a Brownian motion, then the Ito integral $\int_0^t X_s B_s$ is defined by first finding a sequence of elementary processes $E^n_t$ that converge to $X_t$ in $L^2$, and then defining$$\int_0^t X_s dB_s := L^2-\lim_{n \rightarrow \infty} \int E_s^n dB_s.$$I am curious why a similar construction doesn't work for the Lebesgue integral. That is, if $\{\phi_n\}$ are simple and $\phi_n \rightarrow f$ uniformly, then why can we not define$$\int f d\mu := \lim_{n\rightarrow \infty} \int\phi_n d\mu.$$