Understanding the following expression in an integration

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I cannot understand how the parts of this proof circled in red are obtained.

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  1. This is an inequality, which stems from the fact that the function $y\mapsto|x-y|^{-t}\mathbf 1_{|x-y|\gt1}$ is uniformly bounded by $1$ hence its integral is at most $\mu(\mathbb R^n)$.

  2. This is an equality, which stems from the fact that $\mu$ is the restriction of $\mathcal H^s$ to $E\subseteq F$.