Lebesgue Integral is a Measure

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Lebesgue Integral is a Measure

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Theorem

Let $s,t$ be nonnegative measurable simple functions on $X$ . For $E\in\mathcal{M}$ , define $\nu(E)=\int\limits _{E}s \, d\mu$ Then $\nu$ is a Measure on $\mathcal{M}$ . Also $\int\limits _{X}(s+t) \, d\mu=\int\limits _{X}s \, d\mu+\int\limits _{X}t \, d\mu$

Suppose $f:X\to[0,+\infty]$ is measurable, and $\nu(E)=\int\limits _{E}f \, d\mu \quad (E \in \mathcal{M})$ Then $\nu$ is a measure on $\mathscr{M}$ and $\int\limits _{X}g \, d\nu =\int\limits _{X}g\cdot f \, d\mu$ for every measurable $g$ on $X$ with range in $[0,+\infty]$ .

- The converse of this theorem is the Radon-Nikodym Theorem and as a result is intimately connected. - We call $f$ the density of $\nu$ w.r.t. $\mu$ and by the second identity we can write $d\nu=f\,d\mu$ or $\frac{d\nu}{d\mu}=f$ which is commonly referred to as the Radon-Nikodym Derivative

Linked from

A Summary of MATH 891