# Convergence of measure topology

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?