# probability theory – key identities in the proof of the optional sampling theorem

Let $$( Omega, mathscr {F}, mu)$$ be a $$sigma$$– finite measurement space,
let $$( mathscr {F} _n) _ {n in mathbb {N}}$$ to be a filtration of sub-sigma-algebras
of $$mathscr {F}$$, let $$(u_n) _ {n in mathbb {N}}$$ to be a submartingale
related to $$( mathscr {F} _n) _ {n in mathbb {N}}$$,
and let $$sigma$$ and $$tau$$ be bounded time out (compared to $$( mathscr {F} _n) _ {n in mathbb {N}}$$) such as $$color {red} { sigma leq tau leq N}$$.
The optional sampling theorem states that
$$int (u_ sigma-u_ tau) d mu leq0.$$
One way to prove it is to use key identities
$$color {black} {int (u_ sigma-u_ tau) of mu} = color {blue} { int sum_ {n = sigma (w)} mu (dw)} = color {green} { sum_ {n = 0} ^ {N-1} int (u_n-u_ {n + 1}) 1_ {[sigma=n]cap[tau>n]} d mu}.$$
Black = blue identity is easy. However, I fought with the blue -> green. Although it sounds simple, I have not found a rigorous way to establish that.

My question: Is there a rigorous way to establish the identity blue = green?

Any help / hint is highly appreciated.