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We generalize the respective ``double recurrence'' results of Bourgain and ofthe second author, which established for pairs of $L^{\infty}$ functions on afinite measure space the a.e. convergence of the discrete bilinear ergodicaverages and of the discrete bilinear Hilbert averages defined by invertiblemeasure-preserving point transformations. Our generalizations are set in thecontext of arbitrary sigma-finite measure spaces and take the form of a.e.convergence of such discrete averages, as well as of their continuous variablecounterparts, when these averages are defined by Lebesgue space isometries andact on $L^{p_{1}}\times L^{p_{2}}$ ($ 1<p_{1},p_{2}<\infty $,$p_{1}^{-1}+p_{2}^{-1}<3/2$). In the setting of an arbitrary measure space,this yields the a.e. convergence of these discrete bilinear averages when theyact on $L^{p_{1}}\times L^{p_{2}}$ and are defined by an invertiblemeasure-preserving point transformation.

{Spaces of Infinite Measure and Pointwise Convergence of the Bilinear Hilbert and Ergodic Averages Defined by $L^{p}$-Isometries