## Lebesgue Integration

Proposition 8.2.6 Let $\Omega$ be a measurable set, and let $f: \Omega\rightarrow [0,\infty]$ and $g: \Omega\rightarrow [0,\infty]$ be non-negative measurabl...

Definition Let $A\in \mathfrak{M}_{m\times n}(\mathbb{R})$ be a matrix. We define the column rank of $A$ as dimension of $\langle [A]^1, \ldots, [A]^n \rangl... ## Closed Sets and Limit Points 1 minute read Definition (Closed) A subset$A$of a topological space$X$is said to be closed if the set$X-A$is open. ## Product Topology and Subspace Topology 4 minute read Definition (Product Topology on$X \times Y$) Let$(X,\mathfrak{T}_X)$and$(Y,\mathfrak{T}_Y)$be topological spaces. The product space topology on$X\times...
Definition (Topology) A topology on a set $X$ is a collection $\mathfrak{T}$ having the following properties: (1) $\emptyset$ and $X$ are in $\mathfrak{T}$ (...