Summary of Key Results

| Theorem | Conditions | Conclusion | | ---------------------- | ----------------------------------------- | ---------------------------------------- | --------------- | ------------------------ | | Monotone Convergence | $0 \leq f_n \nearrow f$ | $\lim \int f_n = \int f$ | | Fatou”s Lemma | $f_n \geq 0$ | $\int \liminf f_n \leq \liminf \int f_n$ | | Dominated Convergence | $f_n \to f$, $| f_n | \leq g \in L^1$ | $\lim \int f_n = \int f$ | | Holder’s Inequality | $f \in L^p$, $g \in L^q$, $1/p + 1/q = 1$ | $\|fg\|_1 \leq \|f\|_p \|g\|_q$ | | Minkowski’s Inequality | $f, g \in L^p$ | $\|f + g\|_p \leq \|f\|_p + \|g\|_p$ | | Fubini | $f \in L^1(\mu \times \nu)$ | Iterated integrals equal double integral | | Radon-Nikodym | $\nu \ll \mu$, $\sigma$-finite | $d\nu/d\mu$ exists and is unique a.e. |

:::caution Common Pitfall When applying the dominated convergence theorem, the dominating function $g$ must be integrable ($g \in L^1$), not just bounded. A bounded function on a space of infinite measure need not be integrable.

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