The authors introduce a new concept of sensitivity with respect to initial conditions by means of tools of Ergodic Theory. They consider endomorphisms on a probabilistic Lebesgue space \((X,{\mathcal B}, \mu)\) (with a fixed metric \(d\) on \(X\)) and they assume that the support of \(\mu\) (in symbols, supp \(\mu\)) is not a singleton. Their definition is: The endomorphism \(T\) is said to be \textit{pairwise sensitive} (with respect to initial conditions) if there exists \(\delta >0\) -- a sensitive constant -- such that for \(\mu^{\otimes 2}\)-a.e. \((x,y)\in X\times X\), one can find \(n\geq 0\) with \(d(T^{n}(x),T^{n}(y))\geq \delta\). The authors compute the sensitivity constant in the following way: Assume that \(T\) is pairwise sensitive. Then, (i) there exists \(\delta >0\) such that for \(\mu^{\otimes 2}\)-a.e. \((x,y)\in X\times X\), one can find a sequence \((n_{k})_{k\geq 0}\) with \(d(T^{n_{k}}(x), T^{n_{k}}(y))\geq \delta \) for all \(k\geq 0\), (ii) for \(\mu^{\otimes 2}\)-a.e. \((x,y)\in X\times X\), one has \(\sup_{n\geq 0}d(T^{n}(x), T^{n}(y))\geq \Delta (T)\) (here \(\Delta (T)\) stands for the supremum of all sensitive constants for \(T\)) , and (iii) \(\Delta (T)\leq \) diam supp \(\mu\). Moreover, if \(T\) is weakly mixing, then \(T\) is pairwise sensitive and the equality \(\Delta (T)=\) diam supp\,\(\mu\) holds. The relationship between this notion of sensitivity and the metric entropy is also analyzed. To be precise, the following result is proved: if \(T\) is ergodic and \(\alpha =\{\, P_{1},\dots ,P_{n}\,\}\) is a finite measurable partition of \(X\) such that \(P_{1},\dots ,P_{n}\) are \(\mu\)-continuity sets for \(d\) and the metric entropy of \(T\) with respect to the partition \(\alpha\) is positive, then \(T\) is pairwise sensitive. (A measurable set \(A\) in \(X\) is a \(\mu\)-continuity set if its boundary is a set of measure zero).