Measure Theory

Definition

A measure \(\mu\) on \(X\) is a function \(\mu: \Sigma \to [0,\infty]\) where \(\Sigma\) is a \(\sigma\)-algebra, satisfying: - \(\mu(\emptyset) = 0\) - Countable additivity: \(\mu(\bigcup_i E_i) = \sum_i \mu(E_i)\) for disjoint \(E_i\)

Lebesgue Measure

On \(\mathbb{R}^n\), Lebesgue measure \(\lambda\) satisfies: - Translation invariance: \(\lambda(E + x) = \lambda(E)\) - Countable additivity - \(\lambda([0,1]^n) = 1\)

Measurable Functions

\(f: X \to \overline{\mathbb{R}}\) is measurable if \(f^{-1}((a,\infty]) \in \Sigma\) for all \(a \in \mathbb{R}\).

Integration

\[\int_X f \, d\mu = \int_0^\infty \mu(\{x: f(x) > t\}) \, dt\]

Monotone Convergence Theorem (MCT)

If \(0 \leq f_n \uparrow f\) pointwise, then: \[\lim \int f_n \, d\mu = \int f \, d\mu\]

Dominated Convergence Theorem (DCT)

If \(f_n \to f\) a.e. and \(|f_n| \leq g\) with \(g \in L^1\), then: \[\lim \int f_n \, d\mu = \int f \, d\mu\]

\(L^p\) Spaces

\[\|f\|_p = \left(\int |f|^p \, d\mu\right)^{1/p}, \quad 1 \leq p < \infty\]

• Hölder: \(\|fg\|_1 \leq \|f\|_p \|g\|_q\) where \(\frac{1}{p} + \frac{1}{q} = 1\)
• Minkowski: \(\|f + g\|_p \leq \|f\|_p + \|g\|_p\)

Product Measure

\((X \times Y, \Sigma \otimes \tau, \mu \times \nu)\) with: \[(\mu \times \nu)(A \times B) = \mu(A)\nu(B)\]

Fubini’s Theorem: \(\int_{X \times Y} f \, d(\mu \times \nu) = \int_X \int_Y f \, d\nu \, d\mu\)