Modern Abstract Analysis

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4:53 AM

Section on production measures and integration starts on page 158, Fubini's theorem is formulated as Theorem 8.4 on page 162 and Theorem 8.7 on page 165.

Q: Understanding Lang's proof of product $\sigma$-algebras

We have two $\sigma$-finite measure spaces $(X,\mathcal{M},\mu)$ and $(Y,\mathcal{N},\nu)$ and we let $\mathcal{A}$ and $\mathcal{B}$ denote the rings of sets of finite measure in $\mathcal{M}$ and $\mathcal{N}$ respectively. We also let $\mathcal{A}\times \mathcal{B}$ denote the collection of al...

Q: $(\mu\otimes \nu) (Z)=0$ implies $\nu(Z_x)=0$ for almost all $x$.

Am reading the proof Lemma 8.3 in Lang's real and functional analysis book. The set-up is two $\sigma$-finite measure spaces $(X,\mathcal{M},\mu)$ and $(Y,\mathcal{N},\nu)$ and we have a set $Z\in \mathcal{M}\otimes\mathcal{N}$ with $(\mu\otimes \nu)(Z)=0$. We want to show that this implies $\nu...

real-analysis measure-theory lebesgue-integral almost-everywhere

9 hours later…

Alphie

1:27 PM

Yes I have questions regarding the proof of part 1 Theorem 8.4 on page 162.

Martin Sleziak

2:04 PM

@Alphie I am unlikely to help. (I am not very good in measure theory.)

But maybe if you say what your question is, somebody will respond.

Of course, there is also the main chatroom and the main site.

I will add here at least the corresponding page from Google Books.

Alphie

My question is why do we need to invoke Corollary 5.10 on page 163 to show that f_x is integrable? It seems to me that integrability follows from the definition of the integral since we have an L^1 Cauchy sequence of step maps converging the f_x for almost every x.