421 reputation
19
bio website msemac.redwoods.edu/~darnold/…
location Eureka, California
age
visits member for 1 year, 3 months
seen Apr 18 at 3:34

Jul
9
asked Constructing a complete measure space
Jul
9
comment $f$ measurable, $f=g$ almost everywhere, complete measure space
Very well written. Thank you for the response.
Jul
9
comment $f$ measurable, $f=g$ almost everywhere, complete measure space
Nice example, but there is part of it that I don't understand. If your measure space is not complete, how do you prove that there is a non-measurable set contained in a set of measure zero.
Jul
9
accepted $f$ integrable but $f^2$ not integrable
Jul
9
comment $f$ integrable but $f^2$ not integrable
I'd like to thank folks for these extremely helpful examples. This continues to open doors for me.
Jul
9
asked $f$ measurable, $f=g$ almost everywhere, complete measure space
Jul
8
asked $f$ integrable but $f^2$ not integrable
Jul
8
accepted Show there exists an N such that $n\ge N$ implies $\int|f^+-\phi_n|\,d\mu<\epsilon/2$
Jul
8
comment Show there exists an N such that $n\ge N$ implies $\int|f^+-\phi_n|\,d\mu<\epsilon/2$
I think it does help. Because $\int \varphi_n d\mu \nearrow \int f \, d\mu$, there exists an $N$ such that $n\ge N$ implies that $\int f^+\,d\mu-\epsilon/2<\int \phi_n\,d\mu\le \int f^+\,d\mu$. Then $\int f^+\,d\mu-\int \phi_n\,d\mu<\epsilon/2$ for all $n\ge N$, which is saying the same thing as $\int(f^+-\phi_n)\,d\mu<\epsilon/2$ for all $n\ge N$, which is what I need.
Jul
8
comment Show there exists an N such that $n\ge N$ implies $\int|f^+-\phi_n|\,d\mu<\epsilon/2$
At this point (Chapter 5) in Bartle's Elements of Integration, if $X$ is a nonempty set, $\mathcal{X}$ is the $\sigma$-algebra of subsets of $X$, and $\mu$ is a measure defined on $\mathcal{X}$ (i.e., $\mu(E)\ge 0$ for any $E\in\mathcal{X}$, $\mu(\phi)=0$, and $\mu$ is countably additive). If $f\in L(X,\mathcal{X},\mu)$, that means that $f:X\to R$, $f$ is measurable, and $\int f^+\,d\mu<0$ and $\int f^-\,d\mu<0$. The integral of $f$ is defined as $\int f\,d\mu=\int f^+\,d\mu-\int f^-\,d\mu$. The space $L^p$ has not been discussed as yet.
Jul
7
asked Show there exists an N such that $n\ge N$ implies $\int|f^+-\phi_n|\,d\mu<\epsilon/2$
Jul
7
awarded  Teacher
Jul
7
comment $f$ integrable, $g$ measurable, $f = g$ almost everywhere implies $g$ integrable
Hope this is a good proof. I would appreciate any comments.
Jul
7
revised $f$ integrable, $g$ measurable, $f = g$ almost everywhere implies $g$ integrable
deleted 47 characters in body
Jul
7
answered $f$ integrable, $g$ measurable, $f = g$ almost everywhere implies $g$ integrable
Jul
7
comment $f$ integrable, $g$ measurable, $f = g$ almost everywhere implies $g$ integrable
I don't think either hint will work as I can't say that $\int g\,d\mu$ exists until I first show that $\int g^+\,d\mu<+\infty$ and $\int g^-\,d\mu<+\infty$.
Jul
7
asked $f$ integrable, $g$ measurable, $f = g$ almost everywhere implies $g$ integrable
Jul
6
asked The difference of two charges
Jul
2
asked Equating Lebesgue and Riemann integrals
Jun
29
accepted $f, g$ measurable function on $E$ that are finite a.e. on $E$