Kind of passage to the limit in the sense of distributions
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Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$ , and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ satisfies begin{equation} int_{Bsetminus F}f(t)phi(t)dtleq M end{equation} for all test function $phi$ . We know that if we had $$f(x)leq M$$ for all $xin Bsetminus F$ , we could get the same inequality by passing to the limit (supposing that $f$ is continuous); is it possible, in the same way, to circumvent around $F$ in (1) and prove that (1) holds for integrals over $B$ ?
real-analysis integration distribution-theory
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