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Let $\gamma$ denote the Hausdorff/Kuratowski measure of noncompactness defined on a Banach space $(X,\|\cdot\|)$. I was wondering whether $\gamma(A)=\gamma(A+K)$ holds for $A\subset X$ is bounded and $K\subset X$ is compact.

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    Yes, since $\gamma(A) \le \gamma(A + K)$ by monotonicity and $\gamma(A+K) \le \gamma(A) + \gamma(K) = \gamma(A)$ as for example mentioned in the wiki article you linked.2012-05-15

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