  We theoretically investigate magnetic properties of a trapped ultracold Fermi
gas. Including pairing fluctuations within the framework of an extended
$T$-matrix approximation (ETMA), as well as effects of a harmonic trap in the
local density approximation (LDA), we calculate the local spin susceptibility
$\chi_{\rm t}(r,T)$ in the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein
condensation) crossover region. We show that pairing fluctuations cause
non-monotonic temperature dependence of $\chi_{\rm t}(r,T)$. Although this
behavior looks similar to the spin-gap phenomenon associated with pairing
fluctuations in a {\it uniform} Fermi gas, the trapped case is found to also be
influenced by the temperature-dependent density profile, in addition to pairing
fluctuations. We demonstrate how to remove this extrinsic effect from
$\chi_{\rm t}(r,T)$, to study the interesting spin-gap phenomenon purely
originating from pairing fluctuations. Since experiments in cold atom physics
are always done in a trap, our results would be useful for the assessment of
preformed pair scenario, from the viewpoint of spin-gap phenomenon.
