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This allows to find the largest subspace $\\mathcal R_\\mu$ of $\\mathcal L^1(\\Omega)+\\mathcal L^\\infty(\\Omega)$ such that the ergodic averages $\\frac1n\\sum\\limits_{k=0}^{n-1}T^k(f)$ converge almost uniformly (in Egorov's sense) for every $f\\in\\mathcal R_\\mu$ and every Dunford-Schwartz operator $T$. 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