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We study strongly exposed points in general Köthe-Bochner Banach spaces X(E). We first give a characterization of strongly exposed points of the set of X-selections of a measurable multifunction Γ. We then apply this result to the study of strongly exposed points of the closed unit ball of X(E). Precisely we show that if an element f is a strongly exposed point of $B_{X\left(E\right)}$, then |f| is a strongly exposed point of $B_X$ and f(ω)/∥ f(ω)∥ is a strongly exposed point of $B_E$ for μ-almost all ω ∈ S(f).