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Let      ( Ω  , Σ  , P ) (\Omega ,\,\Sigma ,\,P)       be a complete probability space. Let      ( Σ  t ) t ∈  J {({\Sigma _t})_{t \in J}}       be a directed family of sub-     σ  \sigma      -algebras of      Σ  \Sigma      . Let      ( Φ  , Ψ  ) (\Phi ,\,\Psi )       be a pair of conjugate Young functions. We investigate the covering conditions that are equivalent to the essential convergence of      L Ψ  {L^\Psi }      -bounded martingales. We do not assume that either      Φ  \Phi       or      Ψ  \Psi       satisfy the      Δ  2 {\Delta _2}       condition. We show that when      Φ  \Phi       satisfies condition Exp, that is when there exists an      a &gt; 0 a &gt; 0       such that      Φ  ( u ) ≤  exp a u \Phi (u) \leq \operatorname {exp} \,au       for each      u ≥  0 u \ge 0      , the essential convergence of      L Ψ  {L^\Psi }      -bounded martingales is equivalent to the classical covering condition      V Φ  {V_\Phi }      . This covers in particular the classical case      Ψ  ( t ) = t ( log t ) + \Psi (t) = t{(\operatorname {log} \,t)^ + }      . The growth condition Exp on      Φ  \Phi       cannot be relaxed. When      J J       contains a countable cofinite set, we show that the essential convergence of      L Ψ  {L^\Psi }      -bounded martingales is equivalent to a covering condition      D Φ  {D_\Phi }       (that is weaker than      V Φ  {V_\Phi }      ). When      Φ  \Phi       fails condition Exp, condition      D Φ  {D_\Phi }       is optimal. Roughly speaking, in the case of      L 1 {L^1 }      -bounded martingales, condition      D Φ  {D_\Phi }       means that, locally, the Vitali condition with finite overlap holds. We also investigate the case where      J J       does not contain a countable cofinal set and      Φ  \Phi       fails condition Exp. In this case, it seems impossible to characterize the essential convergence of      L Ψ  {L^\Psi }      -bounded martingales by a covering condition. Using the Continuum Hypothesis, we also produce an example where all equi-integrable      L 1 {L^1 }      -bounded martingales, but not all      L 1 {L^1 }      -bounded martingales, converge essentially. Similar results are also established in the derivation setting.

Derivation, 𝐿^{Ψ}-bounded martingales and covering conditions