Cfrgtkky

1 1 $begingroup$ We say that a measurable function $g:mathbb R to mathbb R$ has a Lebesgue point at $x in mathbb R$ if $$ frac1{2s}int_{x-s}^{x+s} |g(y)-g(x)| , dy to 0 quad text{as } s to 0.$$ If $g$ is continuous, then every $x in mathbb R$ is a Lebesgue point of $g$ . The Lebesgue differentiation theorem states that a (locally) Lebesgue integrable function has Lebesgue points almost everywhere. The function given by $$ g(x) = sum_{k=0}^infty (-1)^kchi_{(2^{-k-1},2^{-k}]} $$ is an example of a function that does not have a Lebesgue point at $0$ . Now suppose that $g: mathbb R to mathbb R$ and $h: mathbb R to mathbb R$ have Lebesgue points at every $x in mathbb R$ . Is it possible that the product $gh$ has non-Lebesgue points? real-analysis measure