Theory of integration with respect to a function, continuous martingales I am currently doing courses in Financial Derivatives and in Continuous Martingales and Stochastic calculus, the former being not very rigorous in comparison with the latter, so this is going to be a question on how the stochastic calculus is developed.
For me it's not very clear how things work at the moment so I'm going to share with you what I know and what I'd like to understand.
We define the variation of a function $a$ over $[0,T]$ to be
$$V(a)_T=\sup_\pi \sum_{i=1}^{N(\pi)-1}|a_{t_{i+1}}-a_{t_i}|$$
where $\pi$ is a partition of $[0,T]$ consisting of the points $a_{t_i}$
Then my lecture notes say that we can define measures $\mu_+$ and $\mu_-$ such that
$$\mu_+((0,t])=\frac{V(a)_t+a(t)}{2}, \mu_-((0,t])=\frac{V(a)_t-a(t)}{2}$$ 
and say that 
$$\int_0^tf(s)da(s)=\int_0^tf(s)\mu_+(ds)-\int_0^tf(s)\mu_-(ds)$$
provided we have 
$$\int_0^t|f(s)|(\mu_+(ds)+\mu_-(ds))<\infty$$
The definition I have in the less formal course for integration against a Brownian motion $W_t$ is the following:
$$\int_0^tf(Wu,u)dW_u=\lim_{\pi \rightarrow 0}\sum f(W_k, t_k)(W_{k+1}-W_k)$$
I have the following problems:
$1:$ Shouldn't there be a limit as the mesh of the partition goes to $0$ in the definition of the variance of the function $a$?
$2:$ In the definition of the $2$ measures, we have only information about intervals of the form $(0,t]$. What happens otherwise?
$3:$ Finally, if I am supposed to calculate the Stratonovich integral for example, it involves a limit as the mesh goes to $0$ which we are not given in the rigorous definition.
Sorry if those questions are way too obvious, I just can't get the concept.
Thanks in advance!
 A: $1$: You are confusing the concepts of $p$-variation (in the case $p=1$) and total variation. The object you have defined is the latter and shouldn't include the limit as the mesh size goes to $0$. 
$2$: Notice that the sets of the form $(0,t]$ are enough to define the measure since, by finite additivity, they are enough to specify what happens on sets of the form $(s,t]$ and sets of this form are a $\pi$-system generating the Borel $\sigma$-algebra. 
$3$: To compute the Stratonovich integral of B.M. against itself according to your definition note that
\begin{align*}
\lim_{\delta \to 0} \sum_{j=0}^{N(\pi)-1} \frac{1}{2}(B_{t_{j+1}}+B_{t_j})(B_{t_{j+1}}-B_{t_j}) = \lim_{\delta \to 0} \bigg \{ \sum_{j=0}^{N(\pi)-1} \frac{1}{2} (B_{t_{j+1}}-B_{t_j})^2 + \sum_{j=0}^{N(\pi)-1} B_{t_j}(B_{t_{j+1}}-B_{t_j}) \bigg \}
\end{align*}
where the first sum converges to half of the quadratic variation of Brownian motion and the second to the Ito integral of B.M. against itself. To calculate the latter, you could  for example apply Ito's formula to $B_t^2$. Here both limits are in probability and follow from fairly standard results for both objects. 
