Analysis
Squeeze Theorem
When two bounding sequences converge to the same limit, any sequence caught between them must follow
Monotone Convergence Theorem
A bounded monotone sequence of reals always converges — the supremum is the limit
Cauchy Criterion
A sequence converges if and only if its terms eventually become arbitrarily close to each other — no candidate limit required
Monotone Class Theorem
A pi-system closed under intersection generates the same sigma-algebra as the lambda-system it generates
Fatou's Lemma
The integral of the liminf of nonnegative measurable functions is at most the liminf of their integrals
Dominated Convergence Theorem
Pointwise convergence plus a uniform integrable dominating bound lets you pass the limit inside the integral
Fubini's Theorem
For integrable functions on a product measure space the iterated integrals in either order both equal the double integral
Lebesgue Differentiation Theorem
For locally integrable f, the ball averages of f converge to f(x) at almost every point x
Absolutely Continuous Functions
A function is absolutely continuous iff it is the integral of its derivative — the bridge between differentiation and Lebesgue integration
Arzelà–Ascoli Theorem
A family of continuous functions on a compact space is precompact iff it is equicontinuous and uniformly bounded