When introducing unilateral and bilateral integral transforms (in the context of Laplace transform), I have problems understanding how causality and non-causality are expressed when considering the (finite) discrete case using DFT or Z-Transformation.
In general, bilateral transformations are said to be used for the application of non-causal systems
$$F(s) = \int_{-\infty}^\infty f(t) e^{-st} \, dt, \quad \stackrel{discrete}{\longrightarrow} \quad F(s)= \sum_{n=-\infty}^{\infty} f[n]\ e^{-s T}. $$
Similarly, unilateral transformations are said to be used for causal systems (restricting the integral limits to positive values)
$$F(s) = \int_{0}^\infty f(t) e^{-st} \, dt, \quad \stackrel{discrete}{\longrightarrow} \quad F(s)= \sum_{n=0}^{\infty} f[n]\ e^{-s T}.$$
One may use the convolution operator to introduce the concepts
$$ \text{Causal:} \qquad y \left( t \right) = \int_0^\infty f(\tau) g(t - \tau)\, d\tau,$$
$$ \text{Non-causal:} \qquad y(t)=\int_{-\infty}^\infty f(\tau) g(t - \tau)\, d\tau,$$
$$\text{Anti-causal:} \qquad y(t) =\int _0^\infty f(\tau) g(t + \tau)\, d\tau.$$
It would be nice if someone can give a simple explanation what it means to speak of causality by merely restricting the integral limits and how this is understood in the specific case applying the DFT or Z-transform to a given (finite) dataset/function $f(t), f[n]$.