Convergence integral problem

[aaaa202]

Homework Statement

I have the attached file as an exercise for class. Problem is that I don't really understand why my book spends so much into solving it, when for me it seems pretty easy.

Homework Equations

Lebesgue dominated convergence theorem

The Attempt at a Solution

I thought that since the integrals of g and G are finite they are integrable. Also f_k is integrable for all k. Since f_k is squeezed between g_k and G_k we must have that either lfl≤lGl, which is integrable
or
lfl≤lgl
So lfl is no matter what bounded by an integrable function and we can use the Lebesgue dominated convergence theorem. Why is this not a correct approach?
Is it because finite integral does not imply integrability? I don't see why it wouldn't.

 

[mighty2000]

it is important to carefully analyze and understand problems before jumping to conclusions. In this case, it is important to understand the definitions and assumptions of the Lebesgue dominated convergence theorem before claiming that it is applicable in this situation.

Firstly, the Lebesgue dominated convergence theorem states that if a sequence of functions {f_k} converges pointwise to a function f, and if there exists an integrable function g such that |f_k|≤g for all k, then f is also integrable and the integral of f_k converges to the integral of f.

In this exercise, the assumption is that the integrals of g and G are finite, not that they are integrable. Just because a function has a finite integral does not necessarily mean it is integrable. For example, the function f(x)=1/x on the interval [0,1] has a finite integral, but it is not integrable according to the Lebesgue integral.

Additionally, the Lebesgue dominated convergence theorem requires that the sequence of functions converges pointwise. In this exercise, it is not clear if the sequence of functions {f_k} actually converges pointwise. Even if it does, it is not clear if it converges to the function f in a way that satisfies the assumptions of the theorem.

In summary, while your approach may seem intuitive, it is not a valid application of the Lebesgue dominated convergence theorem. It is important to carefully consider the definitions and assumptions of the theorem before applying it to a problem.