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$\lim_{x'\rightarrow x} f(x') \stackrel{?}{=} f(x)$ for almost all $x$ if $f\in L^1(R^d)$?Pointwise limit at Lebesgue's pointDear MOs, I am sorry if this problem is too elementary for someone. I just want to get confirmation. Suppose $f\in L^1(R^d)$. Since almost all points are Lebesgue points by The the Lebesgue differentiation theorem, can we say that for almost every $x\in R^d$, $$ \lim_{x'\rightarrow x} f(x') = f(x)\:? $$ I think it is true probably only for $d=1$. Does anyone know some results about this problem? Thanks a lot! |
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Dear MOs, I am sorry if this problem is too elementary for someone. I just want to get confirmation. Suppose $f\in L^1(R^d)$. Can Since almost all points are Lebesgue points by The Lebesgue differentiation theorem, can we say that for almost every $x\in R^d$, $$ \lim_{x'\rightarrow x} f(x') = f(x)\:? $$ I think it is true since almost all points are Lebesgue points by The Lebesgue differentiation theorem.probably only for $d=1$. Does anyone know some results about this problem? Thanks a lot! |
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$\lim_{x'\rightarrow x} f(x') \stackrel{?}{=} f(x)$ for almost all $x$ if $f\in L^1(R^d)$?Dear MOs, I am sorry if this problem is too elementary for someone. I just want to get confirmation. Suppose $f\in L^1(R^d)$. Can we say that for almost every $x\in R^d$, $$ \lim_{x'\rightarrow x} f(x') = f(x)\:? $$ I think it is true since almost all points are Lebesgue points by The Lebesgue differentiation theorem. Thanks a lot!
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