The problem

For a smooth function $g(t)$, the Riemann-Stieltjes integral

$$ \int_0^T f(t)\,dg(t) $$

does not depend on whether the Riemann sum uses left endpoints, right endpoints, or midpoints.

What changes for Brownian motion

If we try to define

$$ \int_0^T B_t\,dB_t $$

by sums of the form

$$ \sum_{i=0}^{N-1} B_{\xi_i}\bigl(B_{t_{i+1}}-B_{t_i}\bigr), $$

then different choices of $\xi_i$ lead to different limits.

Consequence

For Brownian motion, stochastic integration is not automatic. We must choose a specific discretization rule.