Consider a sequence of measurable functions $\{f_n\}\rightarrow f,$ almost everywhere $x$ on a finite measurable set $E$.
My questions:
- When we say the sequence is uniformly bounded, does this mean:
$$\exists M\in\mathbb{N},f_n(x)\leq M\space\forall n,x$$
To check (1), we fix $x_0$, see if $f_n$ is bounded by $M$. Do this for all $x\in E$. Is this correct?
When we say $f$ dominates the sequence, does this mean:
$$\forall n,x\space f_n(x)\leq f(x)$$
Also to check (3), we fix $x_0$, see if the entire sequence is bounded by the value $f(x_0).$ Do this for all $x\in E$. Is this procedure correct?
I am studying these concepts in the context of Fatou's Lemma, Monotone and Dominated Convergence Theorem. Why is this boundedness important in converence theorems? The way I understand is through the Dirac delta function. This function behaves in the limit such a way when the function takes a materially large value, the domain on which such this happens is a negligible set. So does the boundedness or dominance take care of this issue? Is this the correct intuition?