# Lebesgue Integrals of Not-Necessarily Positive Functions

I'm trying to work through the following problem.

Let $$\{f_{n}\}_{n\geq 0}$$ be a sequence of $$L^{1}(\mu)$$ functions (not necessarily positive) for some positive measure $$\mu$$ defined on a measurable space $$(X,\mathfrak{M})$$. Suppose that $$f_{n}(x)\leq f_{n+1}(x)$$ for all $$x\in X$$ and all $$n\geq 0$$, and that $$\lim_{n\to\infty}f_{n}(x)=f(x)$$, where $$f\in L^{1}(\mu)$$. Is it true that $$\int_{X}f_{n}~d\mu\to\int_{X}f~d\mu$$ as $$n\to\infty$$?

My thoughts: I'm not sure if my intuition is correct, but I do not think that the sequence of integrals $$\int_{X}f_{n}~d\mu$$ converges to $$\int_{X}f~d\mu$$. If I'm not mistaken, the given sequence should satisfy all but one hypothesis of the Monotone Convergence Theorem (namely that they are not assumed to be positive functions). I'm thinking that there's some counterexample, and was trying to play around with the definitions, but I'm pretty much stuck. Any help is appreciated!

• this might help (the dominating function is this case would be $|f|$) mathworld.wolfram.com/LebesguesDominatedConvergenceTheorem.html – Pink Panther Sep 23 '18 at 16:00
• @PinkPanther Okay, I think that makes sense (I hadn't even thought of the Dominated Convergence Theorem). I am a little confused as to why the dominating function is $|f|$. – Sir_Math_Cat Sep 23 '18 at 16:05
• I think dominating function needs to be something slightly more clever than $|f|$. Perhaps $|f_1|+|f|$ would work. – Aweygan Sep 23 '18 at 16:12
• What if you used MCT on $g_n:=f_n-f_1$, which is a monotone sequence of positive integrable functions? – John Dawkins Sep 23 '18 at 17:09
• @JohnDawkins That looks like a pretty good idea. Shouldn't it be $g_{n}=f_{n}-f_{0}$ though? – Sir_Math_Cat Sep 23 '18 at 17:14