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Let {(Xi, Si, μi) : i ∈N} be a sequence of probability measure spaces and (↑*Xi, L(↑*Si), L(↑*μi)) be the Loeb measure space with respect to (Xi, Si, μi)for i∈N.lET x=↑∞×i=1 Xi,S=↑∞×i=1 Si,μ=↑∞×i=1 μi,We prove that ↑∞×i=1 L(↑*Si)（∪→）L1