The author presents a definition of seminormality of functions taking values in \({\bar {\mathbb{R}}}\), extending classical notions due to Tonelli, McShane, Cesari, and states that the notion of seminormality in the small coincides with seminormality in the large, i.e. seminormality of the integrand \(\ell:T\times X\times V\to {\bar {\mathbb{R}}}\) where X is a Suslin metric space, V a locally convex Suslin space paired with a locally convex Suslin space W by \(\), is equivalent to seminormality of the integral functional \(I_{\ell}: {\mathcal X}\times {\mathcal V}\to {\bar {\mathbb{R}}}\) where \({\mathcal X}\) is the decomposable set of equivalence classes of (\({\mathcal F},{\mathcal B}(X))\)-measurable functions from T to X, equipped with the essential supremum metric, and (\({\mathcal V},{\mathcal W})\) is a pair of decomposable vector spaces of equivalence classes of scalarly \(\mu\)- integrable functions from T to V and W respectively, paired by \(=\int_{T}\), and \(I_{\ell}\) is given by the formula \(I_{\ell}(v,w)=\int_{T}\ell (.,v(.),w(.))\), where (v,w)\(\in {\mathcal V}\times {\mathcal W}.\) By introducing the notion called Nagumo tightness (grosso modo, the notion of inf compactness of the integrand), the author shows that the usual sufficient conditions for the lower semicontinuity of the integral functional of type \(I_{\ell}\), associated to the integrand \(\ell\), imply seminormality of \(I_{\ell}\). Sufficient conditions for sequential lower semicontinuity of \(I_{\ell}\) are also presented in a general form under the Nagumo tightness condition. The reviewer observes that in some usual cases, Nagumo tightness in theorem 4.9 for \((v_ k)\) is equivalent to the hypothesis of the existence of a measurable multifunction \(\Gamma\) with compact convex such that \(v_ k(t)\in \Gamma (t)..\). on \(A_{\epsilon}\) etc..., given in Corollary 4.10; however, this equivalence is not obvious.