Let f be a positive measurable function and $\varphi_k$ a monotone sequence of positive step functions such that $\varphi_k \leq \varphi_{k+1}$ and $\varphi_k \rightarrow f$ pointwise. Then we define the integral of f as follows:
$\int_D f d\mu:= lim_{k \rightarrow \infty} \int_D \varphi_k$.
Now my question is why is the definition of the integral unique? How can I show, that I get for a sequence of step functions $\psi_k$ with the same properties as $ \varphi_k$ the same integral?
The usual approach is not to define the integral exactly that way. Instead define $\int f$ to be the sup of $\int\phi$ over simple functions $\phi$ with $0\le\phi\le f$. Then prove the Monotone Convergence Theorem and it follows that this definition is equivalent to the problematic "definition" you propose.