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As shown in \\cite{Kuksin2004} the noise scaling $\\sqrt{{\\nu}}$ is the only one which leads to non-trivial limiting measures, which are invariant for the 2D Euler equations. We show that any limiting measure $\\mu_{0}$ is in fact supported on bounded vorticities. Relationships of $\\mu_{0}$ to the long term dynamics of Euler in the $L^{\\infty}$ with the weak$^{*}$ topology are discussed. 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