Summary: Let f be a weakly measurable function with values in the Banach space X, defined on the finite measure space (\(\Omega\),\(\Sigma\),\(\mu)\), and \(\lambda =\mu f^{-1}\) its distribution induced on the Baire subsets of (X,\(\sigma\) (X,X’)). It is proved that f is Pettis integrable if and only if \(\lambda\) has the following property: Whenever a net \(Z_ j\) of convex zero sets decreases to the empty set, \(\lambda (Z_ j)\to 0\). This gives us Baire measure characterizations of the Banach spaces with the Pettis integrability property and the Pettis-essentially separable Banach spaces.