Understanding Limit theorems in Measure-theoretic probability theory I have introduced with some of limit theorems like:

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*Bounded convergence theorem: Suppose that $|X_n|\leq M\textrm{ a.s.}$ and $X_n\stackrel{\mathbb P}{\longrightarrow}X$. Then, $\mathbb E\:X=\lim_{n\rightarrow\infty}\mathbb E\:X_n$

*Fatou's Lemma: If $X_n\geq0\textrm{ a.s.}$ for all $n$, $\lim\inf_{n\rightarrow\infty}\mathbb E\:X_n\geq\mathbb E(\lim\inf_{n\rightarrow\infty}X_n)$

*Monotone Convergence Theorem: If $X_n\geq0\textrm{ a.s.}$ and $X_n\uparrow X\textrm{ a.s.}$ then $\mathbb E\:X_n\uparrow\mathbb E\:X$

*Dominated Convergence Theorem: If $X_n\rightarrow X\textrm{ a.s.}$ and $|X_n|\leq Y\textrm{ a.s.}$ for some integrable $Y$, then $\mathbb E\:X=\lim_{n\rightarrow\infty}\mathbb E\:X_n$
I manage to understand their proof but can't understand when and how to apply them and what they express. What is the relation (if any) between them? Which one is more strict (and in which situation)? Is there an easy way to understand what they express (kind of like drawing a triangle for the triangle inequality)? What is a typical application in probability theory?
 A: Fatou's Lemma, the Monotone Convergence Theorem (MCT), and the Dominated Convergence Theorem (DCT) are three major results which answer the question "When do $\lim_{n\rightarrow\infty}$ and $\int$ commute?"
Same definition in measure theory:

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*Monotone Convergence Theorem: If $\{f_n:X\rightarrow[0,\infty)\}$ is a sequence of measurable functions on a measurable set $X$ such that $f_n\rightarrow f$ pointwise almost everywhere and $f_1\leq f_2\leq\cdots,$ then $$\lim_{n\rightarrow\infty}\int_{X}f_n=\int_{X}f$$


*Dominated Convergence Theorem: If $\{f_n:\mathbb R\rightarrow\mathbb R\}$ is a sequence of measurable functions which converge pointwise almost everywhere to $f$, and if there exists an integrable function $g$ such that $|f_n(x)|\leq g(x)$ for all $n$ and for all $x$, then $f$ is integrable and $$\int_{\mathbb R}f=\lim_{n\rightarrow\infty}\int_{\mathbb R}f_n$$

 The MCT and DCT tell us that if  you place certain restrictions on both the $\int_n$ and $\int$, then you can interchange the limit and integral.


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*Fatou's Lemma: Let $(X,Σ,\mu)$ be a measure space and $\{f_n:X\rightarrow[0,\infty]\}$ a sequence of nonnegative measurable functions, then the function $\lim_{n\rightarrow\infty}f_n$ is measurable and $$\int_{X}\lim_{n\rightarrow\infty}\inf f_n\:d\mu\leq\lim_{n\rightarrow\infty}\inf\int_{X}f_n\:d\mu$$

 On the other hand, Fatou's Lemma says, "Here's the best you can do if you don't put any restrictions on the functions."

Wait a minutes, did you say I am not using expectation, then recall the definition,
In general, if $X$ is a random variable defined on a probability space $(\Omega ,\Sigma ,P)$, then the expected value of $X$, denoted by $\mathbb{E}\:X$, is defined as the Lebesgue integral,
$$\mathbb E\:X=\int_{\Omega}X(\omega)\:dP(\omega)$$
Which is nothing more than integration.
