I have a sequence of functions $(f_n)$ from a measure space $(X, \Sigma, \mu)$ that converge pointwise to an integrable function $f$, and I am interested in showing $$ \lim_{n\to\infty} \int f_n\,\text d\mu = \int f\,\text d\mu $$ via the dominated convergence theorem (DCT), but my problem is that I only have a bound on the limit, i.e. a function $g$ with $g(x) \geq |f(x)|$ for every $x\in X$. I don't have a bound on the individual $f_n$. Can I still use DCT?
My intuition is that because $f_n(x)\to f(x)$ then eventually the $f_n$ are really close to being bounded by $g$, but I'm not sure how to get a function $\tilde g$ that is actually an upper bound for all of the $f_n$ or how to make this intuition rigorous.
I was thinking something along the lines of making a set $A(n, \varepsilon) = \{x \in X : |f_k(x)| \leq g(x) + \varepsilon \text{ for all but finitely many } k \geq n\}$ and looking at $\mu(A(n, \varepsilon))$ but I'm still not sure how to turn that into a proof.
First you'll need your $g$ to be integrable to even think of applying the dominated convergence theorem, and even then this is not generally true. In fact even if $f_n\to f$ uniformly and $f$ is bounded by an integrable function this is not necessarily true.
Just consider the measure space $(\mathbb R,\mathcal B,\lambda)$, and each $f_n:=\frac{1}{2n}\mathbb 1_{[-n,n]}$. Clearly $$\int_\mathbb Rf_n=1$$ for all $n$, but $f_n\to 0$ uniformly.