## Difference between spaces of integrable functions w.r.t Lebesgue measure and Borel measure [closed]

Is there a difference between $L^p(\mathbb R,\mathfrak B,\beta)$ and $L^p(\mathbb R,\mathfrak L,\lambda)$ ? Here I denoted by $\lambda$ the Lebesgue measure, defined on the Lebesgue $\sigma$-algebra $\mathfrak L$ and by $\beta$ its restriction to the Borel $\sigma$-algebra $\beta$. Does the answer depend on wether I consider equivalence classes of functions or not?

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Sounds like a homework assignment... See FAQ – Anthony Quas Jan 11 2012 at 20:28

## closed as too localized by Anthony Quas, Dmitri Pavlov, Alain Valette, Bill Johnson, Gerald EdgarJan 12 2012 at 3:35

The $L^p$ spaces however, constist both of equivalence classes of functions, and in fact the spaces are isomorphic via a natural embedding from the Borel one to the other. The difference is that the equivalence classes are bigger. You get more measurable functions in the $\mathcal{L}^p(\lambda)$ space since the sigma-algebra is bigger, but you factor out those you got more when descending to $L^p(\lambda)$.
I thought the Baire algebra was the sigma-algebra generated by the zero sets. $\;$ – Ricky Demer Jan 11 2012 at 20:39