Blog
Notes on measure theory, functional analysis, and geometry.
Picard-Lindelöf Theorem
We discuss the Picard-Lindelöf theorem on local existence and uniqueness of solutions to first-order ODEs, building up from the completeness of C(K,E) under the supremum norm, the Banach fixed point theorem, and Grönwall's inequality.
Fatou's Lemma and applications
We discuss Fatou's lemma, a fundamental inequality that allows interchange of limits and integrals for nonnegative measurable functions, together with its classical applications to MCT, DCT and completeness of Lp space.
Linear Transformations and the Scaling of Lebesgue Measure
How the determinant controls volume distortion, and why translation invariance uniquely characterizes Lebesgue measure up to a scalar.
Surface area and volume of a sphere in finite Euclidean space
A derivation of the volume of the Euclidean unit ball and the surface area of the unit sphere via spherical coordinates, induced metrics, and the Gaussian integral.
Proof of the Arzelà–Ascoli Theorem
Equicontinuity, uniform boundedness, and why compactness in function spaces requires more than pointwise control.
Vitali Covering Theorem
A proof of the Vitali covering theorem via Zorn's lemma and a 5B covering argument.