Uniform Bound, Dominance, and Convergence Theorems

Question

Consider a sequence of measurable functions $\{f_n\}\rightarrow f,$ almost everywhere $x$ on a finite measurable set $E$.

My questions:

1. When we say the sequence is uniformly bounded, does this mean:

$$\exists M\in\mathbb{N},f_n(x)\leq M\space\forall n,x$$

2. 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?

3. When we say $f$ dominates the sequence, does this mean:

$$\forall n,x\space f_n(x)\leq f(x)$$

4. 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?

5. 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?

Answer 1

1. There is no particular reason to require $M \in \Bbb N$. Any $M \in \Bbb R$ will do. Of course, that make no change in the functions meeting the condition, since if you have a negative or non-integer bound, any higher positive integer bound will also work. Also, the condition you give is for a uniform upper bound. In general, uniformly bounded means bounded above and below: $f_n$ is uniformly bounded if $$\exists M \in \Bbb R, \forall x\in E, \forall n \in\Bbb N, |f_n(x)| \le M$$ In measure theory, one generally isn't worried about the behavior of the functions on sets of measure $0$, so there is also the notion of "uniformly bounded almost everywhere": $$\exists E_0 \subset E, \mu(E \setminus E_0) = 0 \text{ and }\exists M \in \Bbb R, \forall x\in E_0, \forall n \in\Bbb N, |f_n(x)| \le M$$
2. If you have an uncountably infinite amount of time and paper, sure, you can do it that way. More practical approaches use the defining properties of the sequence $f_n$ to show that the condition holds on larger subsets of $E$ than single points.
3. When $f_n$ consists of positive functions, that is true, but generally it means that $\forall x, |f_n(x)| \le f(x)$. Note that the dominating function need not be the limit of the sequence $f$. If $|f_n| \le g$ for any integrable function $g$, then the Dominated Convergence Theorem applies.
4. See 2.
5. The "Dirac delta function" is not a function as described in these theorems. The theorems do not apply to the various sequences that "converge" to it. In fact, those sequences converge to the $0$ function except for a single point, where they converge to $+\infty$. The integral of that function, of any other function multiplied by it, will always be $0$. The sequences are not bounded, nor dominated by any integrable function.

To properly discuss the Dirac delta function in mathematics, there are two approaches:

• Consider the "function" to be nothing more than a notational device, so $\int f(x)\delta(x - a)\, dx$ is just a really funky way of writing $f(a)$.
• Generalize the concept of "function" to be more than just maps from a set into $\Bbb R$. When you do this, you must also generalize the concept of integration to cover these new functions (as well as every other operation one wishes to apply to these new "functions"). And you must prove new theorems to deal with them. You cannot simply apply old results such as the DCT, MCT, or Fatou's lemma to them.

Which a particular author uses depends on how deeply they want to get into the subject.