From the point of view of analysis, what is Ito formula?
Is it an integral by substitution, or, a radon-nikodym derivative?
Define the probability space $$ \left(C\left(\Bbb R_+\right),\sigma\left(C\left(\Bbb R_+\right)\right),P\right), $$ where $P$ is the standard Brownian motion measure.
Let $f(x)=x^2$, with Ito formula, I write $$ \int_{C\left(\Bbb R_+\right)} f(X_t)dP(X)=\int_{C\left(\Bbb R_+\right)} \left\{\int_0^t 2X_sdX_s+t \right\} dP\left(X\right). $$ The previous equation is my heuristic to explain Ito formula to an analyst.
This heuristic is itself inspired by the following heuristic which I once heard
Ito formula is a way of expressing how the measure $P$ changes from $X_t$ to $f(X_t)$