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We extend the classical Lyapunov inequality on the measurable space withinfinite measure and on the so-called Grand Lebesgue spaces (GLS).  We find also the exact value for correspondent constant.  Possible applications: Functional Analysis (for instance, interpolation ofoperators), Integral Equations, Probability Theory and Statistics (tailestimations for random variables) etc.

Sharp Lyapunov's inequality for the measurable sets with infinite measure, with generalization to the Grand Lebesgue spaces