How does a Lebesgue measurable set differ from a Borel measurable set?

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I have seen a few of other questions on here that provide examples of sets that are Lebesgue measurable but not Borel measurable, but I believe I have a lapse in understanding somewhere. I know that the Lebesgue measure is a special case of a Borel measure, so then are they not both defined on the sigma algebra generated by all Borel sets?