A question about the outer measure generated
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Let $mathcal{K}subseteq 2^{X}$ be a family of subset such that $emptysetin mathcal{K}$ and let $nucolonmathcal{K}to [0,+infty]$ an application such that $nu(emptyset)=0.$ We denote with $$mathcal{R}_{mathcal{K}}big(Ebig)=bigg{{I_k}_{kinmathbb{N}}subseteqmathcal{K};bigg|;Esubseteqbigcup_{kinmathbb{N}}I_kbigg}.$$ We define $$mu^{(mathcal{K},nu)}big(Ebig):=infbigg{sum_{kinmathbb{N}}nu(I_k);bigg|;{I_k}inmathcal{R}_{mathcal{K}}big(Ebig)bigg}quadtext{if}quadmathcal{R}_{mathcal{K}}big(Ebig)neemptyset$$ and $$mu^{(mathcal{K},nu)}big(Ebig):=+inftyquadtext{if}quad mathcal{R}_{mathcal{K}}big(Ebig)=emptyset.$$ Question. If I know that $mathcal{R}_{mathcal{K}}big(Ebig)ne emptyset$ , can I conclude that $mu^{(mathcal{K},mu)}big(Ebig)<+infty?$ For me the answer is no. Because we do not know that e