Newton’s Method for Solving Systems of Many Nonlinear Equations

FoldUnfold Table of Contents Newton’s Method for Solving Systems of Many Nonlinear Equations Newton’s Method for Solving Systems of Many Nonlinear Equations We will now extend Newton’s Method further to systems of many nonlinear equations. Consider the general system of $n$ linear equations in $n$ unknowns: (1) begin{align} f_1(x_1, x_2, …, x_n) = 0 \ […]

Measures on Algebras of Sets

FoldUnfold Table of Contents Measures on Algebras of Sets Measures on Algebras of Sets Definition: Let $X$ be a set and let $mathcal A$ be an algebra of sets (not necessarily a $sigma$-algebra) on $X$. A Measure on $mathcal A$ is a set function $mu : mathcal A to [0, infty]$ with the following properties: […]

Outer Measurable Sets

FoldUnfold Table of Contents Outer Measurable Sets Outer Measurable Sets Recall from the Outer Measures on Measurable Spaces page that if we have the measurable space $(X, mathcal P(X))$ then an outer measure on this space is a set function $mu^* : mathcal P(X) to [0, infty]$ with the following properties: 1) $mu^*(emptyset) = 0$. […]

Outer Measures on Measurable Spaces

FoldUnfold Table of Contents Outer Measures on Measurable Spaces Outer Measures on Measurable Spaces Definition: Let $(X, mathcal P(X))$ be a measurable space. An Outer Measure on this space is a set function $mu^* : mathcal P(X) to [0, infty]$ with the following properties: 1) $mu^*(emptyset) = 0$. 2) If $A$ and $B$ are subsets […]

The Dominated Convergence Theorem for Measurable Functions

FoldUnfold Table of Contents The Dominated Convergence Theorem for Measurable Functions The Dominated Convergence Theorem for Measurable Functions Recall from The Lebesgue Dominated Convergence Theorem that if $(f_n(x))_{n=1}^{infty}$ is a sequence of Lebesgue measurable functions defined on a Lebesgue measurable set $E$ such that: 1) There exists a nonnegative Lebesgue integrable function $g$ such that […]

The Comparison Test for Integrability

FoldUnfold Table of Contents The Comparison Test for Integrability The Comparison Test for Integrability Recall from The Comparison Test for Lebesgue Integrability that if $f$ is a Lebesgue measurable function defined on a Lebesgue measurable set $E$ and if there exists a nonnegative Lebesgue measurable function $g$ on $E$ such that: 1) $|f(x)| leq g(x)$ […]