7 — Integral Properties, Jensen, Hölder, and the $L^p$ Triangle Inequality
Lecture 7 finishes the fundamental properties of the integral, proves Jensen’s inequality, and derives Hölder’s inequality and the 𝐿 𝑝 L p triangle inequality.
We work on a σ-finite measure space $(\Omega, \mathcal{F}, \mu)$.
A function $f$ is integrable if: \(\int_\Omega \vert f\vert \, d\mu < \infty.\)
Recall: \(\vert f\vert = f^{+} + f^{-}, \quad f^+ = f\vee 0, \quad f^- = -(f\wedge 0),\) and \(I(f) = I(f^+) - I(f^-).\)
1. Basic Properties of the Integral
If $f,g$ are integrable, then:
(1) Monotonicity
\(f \ge g \implies I(f) \ge I(g).\)
(2) Homogeneity
\(I(af) = a\, I(f), \quad a\in\mathbb{R}.\)
(3) Additivity
\(I(f+g) = I(f) + I(g).\)
(4) Absolute value inequality
\(\vert I(f)\vert \le I(\vert f\vert ).\)
Proof (page 1):
\[\vert I(f)\vert = \vert I(f^+) - I(f^-)\vert = \max\{ I(f^+) - I(f^-),\; I(f^-) - I(f^+)\} \le I(f^+) + I(f^-) = I(\vert f\vert ).\]2. Jensen’s Inequality
Assume $\mu(\Omega) = 1$.
Let $\varphi : \mathbb{R} \to \mathbb{R}$ be convex.
Assume $f, \varphi(f)$ are integrable.
To prove: \(\varphi(I(f)) \le I(\varphi(f)).\)
Key fact (supporting line characterization of convexity)
For any convex $\varphi$, for every $x_0\in\mathbb{R}$, there exists an affine map: \(\ell(x) = a + b x\) such that:
- $\ell(x) \le \varphi(x)$ for all $x$,
- $\ell(x_0) = \varphi(x_0)$.
This is the tangent (supporting) line.
Proof
Let $x_0 = I(f)$.
Let $\ell(x) = a + bx$ be the supporting line at $x_0$.
Then for all $x$:
\(\varphi(x) \ge \ell(x).\)
Thus: \(I(\varphi(f)) \ge I(\ell(f)) = \int_\Omega (a + b f)\, d\mu = a + b I(f) = \ell(I(f)) = \varphi(I(f)),\) since $\ell(I(f)) = \varphi(I(f))$ by construction.
Therefore: \(\boxed{\varphi(I(f)) \le I(\varphi(f)).}\)
3. Connection to AM ≥ GM
Take a discrete probability space $\Omega = {1,\dots,n}$,
$\mu({k}) = 1/n$.
Let $f(k) = x_k > 0$.
Apply Jensen to $\varphi(x) = \log x$ (concave; or apply convexity to $-\log x$):
This yields:
\[\log\left(\frac{1}{n}\sum_{k=1}^n x_k\right) \ge \frac{1}{n}\sum_{k=1}^n \log(x_k) = \log\left( \prod_{k=1}^n x_k^{1/n} \right).\]Exponentiate:
\[\frac{1}{n}\sum_{k=1}^n x_k \ge \left( \prod_{k=1}^n x_k \right)^{1/n}.\]This is the classical AM–GM inequality.
4. Young’s Inequality (Used for Hölder)
Let $p,q \ge 1$ with: \(\frac{1}{p} + \frac{1}{q} = 1.\)
For $x,y > 0$, \(xy \le \frac{x^p}{p} + \frac{y^q}{q}.\)
Proof (page 2):
Take \(x_1 = x^p,\quad x_2 = y^q,\quad \omega_1 = \frac{1}{p},\quad \omega_2 = \frac{1}{q}.\)
Using Jensen’s inequality for the exponential or the convex function $u\mapsto e^u$ or directly via AM–GM on $(x^p)^{1/p}$ and $(y^q)^{1/q}$:
\[x y = (x^p)^{1/p}(y^q)^{1/q} \le \frac{x^p}{p} + \frac{y^q}{q}.\]5. Hölder’s Inequality
Let $f,g$ be measurable. If $\vert f\vert ^p$ and $\vert g\vert ^q$ are integrable, then: \(\int_\Omega \vert f g\vert \, d\mu \le \left( \int_\Omega \vert f\vert ^p\, d\mu \right)^{1/p} \left( \int_\Omega \vert g\vert ^q\, d\mu \right)^{1/q}.\)
Proof outline (page 2)
Normalize: \(F = \frac{\vert f\vert }{\vert f\vert _{L^p}}, \qquad G = \frac{\vert g\vert }{\vert g\vert _{L^q}},\) so that: \(\vert F\vert _{L^p} = 1, \quad \vert G\vert _{L^q} = 1.\)
Apply Young’s inequality pointwise to $F(\omega)$ and $G(\omega)$: \(\vert F G\vert \le \frac{\vert F\vert ^p}{p} + \frac{\vert G\vert ^q}{q}.\)
Integrate: \(\int \vert FG\vert \le \frac{1}{p}\int \vert F\vert ^p + \frac{1}{q}\int \vert G\vert ^q = \frac{1}{p}\cdot 1 + \frac{1}{q}\cdot 1 = 1.\)
Undo the normalization: \(\int \vert fg\vert \le \vert f\vert _{L^p} \vert g\vert _{L^q}.\)
Thus: \(\boxed{ \vert fg\vert _{L^1} \le \vert f\vert _{L^p}\vert g\vert _{L^q}. }\)
6. Triangle Inequality in $L^p$
For $p \ge 1$, the space: \(L^p(\Omega,\mathcal{F},\mu) = \left\{ f : \int \vert f\vert ^p < \infty \right\}\) satisfies: \(\vert f + g\vert _{L^p} \le \vert f\vert _{L^p} + \vert g\vert _{L^p}.\)
Sketch of proof
Start with: \(\vert f+g\vert ^p \le (\vert f\vert +\vert g\vert )^p.\)
Use convexity of $t\mapsto t^p$ or Minkowski’s inequality (derived by applying Hölder to $\vert f+g\vert ^{p-1}$):
\[\int \vert f+g\vert ^p \le \left( \vert f\vert _{L^p} + \vert g\vert _{L^p} \right)^p.\]Taking $p$-th roots:
\[\boxed{ \vert f+g\vert _{L^p} \le \vert f\vert _{L^p} + \vert g\vert _{L^p}. }\]This completes the construction of the $L^p$ norm.
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