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Let      T T       be a bounded linear operator from      L 1 ( G , λ  ) {L_1}(G,\lambda )       into      L 1 ( Ω  , F , P ) {L_1}(\Omega ,\mathcal {F},P)      , where      ( G , λ  ) (G,\lambda )       is a compact abelian metric group with its Haar measure, and      ( Ω  , F , P ) (\Omega ,\mathcal {F},P)       is a probability space. Let      ( μ  ω  ) ({\mu _\omega })       be the random measure on      G G       associated to      T T      ; that is,      T f ( ω  ) = ∫  G f ( t ) d μ  ω  ( t ) Tf(\omega ) = \int _G {f(t)d{\mu _\omega }(t)}       for each      f f       in      L 1 ( G ) {L_1}(G)      . We show that, unlike the ideals of representable and Kalton operators, there is no subideal      B B       of      M ( G ) \mathcal {M}(G)       such that      T T       is completely continuous if and only if      μ  ω  ∈  B {\mu _\omega } \in B       for almost      ω  \omega       in      Ω  \Omega      . We actually exhibit a noncompletely continuous operator      T T       such that      μ  ^  ω  ∈  l 2 + ε  ( G ^  ) {\hat \mu _\omega } \in {l_{2 + \varepsilon }}(\hat G)       for each      ε  &gt; 0 \varepsilon &gt; 0      .

A noncompletely continuous operator on 𝐿₁(𝐺) whose random Fourier transform is in 𝑐₀(𝐺̂)