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Let $(\Omega,\mu)$ be a $\sigma$-finite measure space. Let $1\leq p<\infty.$ Suppose $T:L^p(\Omega)\to L^p(\Omega)$ be a bounded positive function, i.e. it takes nonnegative functions to nonnegative functions. Suppose he restriction of $T$ on any $L^p(A)$, where A has finite measure, is continuous in topology of convergence in measure for both domain and codomain ($L^p(A)$ has the $\sigma$ algebra as the restriction of the whole $\sigma$-algebra). Does it imply that $T:L^p(\Omega)\to L^p(\Omega)$ is continuous in the topology of convergence in measure?

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