A $p$-integrable function is a function $f$ defined on a measurable space (typically a subset of $\mathbb{R}^n$) for which the $p$-th power of the absolute value is integrable. Specifically, if $f$ is a measurable function on a domain $\Omega \subset \mathbb{R}^n$, then $f$ is said to be $p$-integrable if the following integral is finite: $ \int_{\Omega} |f(x)|^p \, dx < \infty $ where $1 \leq p < \infty$. ## Explanation of $p$-Integrability ### 1. The $L^p$ Space The collection of all $p$-integrable functions on $\Omega$ forms a space called the $L^p$ space, denoted $L^p(\Omega)$. This space is defined by $ L^p(\Omega) = \{ f : \Omega \to \mathbb{R} \text{ (or } \mathbb{C} \text{)} \mid \int_{\Omega} |f(x)|^p \, dx < \infty \}. $ An $L^p$ space is a Banach space, where the norm is given by $ ||f||_{L^p} = \left( \int_{\Omega} |f(x)|^p \, dx \right)^{\frac{1}{p}}. $ ### 2. Common Cases of $p$: - **$L^1$ (Absolutely integrable functions):** When $p = 1$, $f$ is integrable in the traditional sense: $ \int_{\Omega} |f(x)| \, dx < \infty. $ These are functions for which the total “area under the curve” is finite. - **$L^2$ (Square-integrable functions):** When $p = 2$, $f$ is square-integrable: $ \int_{\Omega} |f(x)|^2 \, dx < \infty. $ $L^2$ functions are especially important in applications involving inner product spaces, like quantum mechanics and signal processing. - **$L^\infty$ (Essentially bounded functions):** For $p = \infty$, $f$ is not typically considered $p$-integrable, but instead is called essentially bounded, meaning that there exists some bound $M$ such that $|f(x)| \leq M$ for almost all $x \in \Omega$. This bound is captured by the essential supremum norm: $ ||f||_{L^\infty} = \operatorname{ess\ sup}_{x \in \Omega} |f(x)|. $ ## Practical Meaning The concept of $p$-integrability allows for the classification of functions based on their “growth” or “magnitude” and determines how they behave under integration. For example, higher values of $p$ in $L^p$ spaces tend to “penalize” larger values of $|f(x)|$ more, providing different tools for analyzing functions across applications in functional analysis, probability, and differential equations. ## Questions ### How to prove something is $p$-integrable? To determine whether a function $f$ is $p$-integrable over a domain $\Omega$, you need to assess whether the integral of its $p$-th power is finite: $ \int_{\Omega} |f(x)|^p \, dx < \infty. $ #### Proving $p$-Integrability ##### 1. Direct Computation - **Compute the Integral Explicitly**: If possible, compute $\int_{\Omega} |f(x)|^p \, dx$ directly. - **Example**: For $f(x) = e^{-x}$ on $[0, \infty)$ and $p = 2$, $ \int_{0}^{\infty} |e^{-x}|^2 \, dx = \int_{0}^{\infty} e^{-2x} \, dx = \frac{1}{2} < \infty. $ Thus, $f$ is 2-integrable on $[0, \infty)$. ##### 2. Use of Inequalities and Comparison Theorems - **Comparison Test**: Compare $|f(x)|^p$ with another function $g(x)$ whose integral is known. - **Dominated Convergence Theorem**: If $|f_n(x)| \leq g(x)$ for all $n$ and $\int |g(x)|^p \, dx < \infty$, then $f_n$ converges in $L^p$. - **Example**: Show $f(x) = \frac{1}{x^\alpha}$ is $p$-integrable on $[1, \infty)$ if $\alpha p > 1$. - Compare $|f(x)|^p = x^{-\alpha p}$ with $x^{-\beta}$, where $\beta = \alpha p$. - Since $\int_{1}^{\infty} x^{-\beta} \, dx$ converges if $\beta > 1$, $f$ is $p$-integrable if $\alpha p > 1$. ##### 3. Estimation Techniques - **Bounding the Integrand**: Find upper bounds for $|f(x)|^p$ that are easier to integrate. - **Use of Known Integrals and Tables**: Utilize standard integrals from calculus to evaluate or estimate the integral. #### Proving Non-$p$-Integrability ##### 1. Show the Integral Diverges - **Compute or Estimate the Integral**: Demonstrate that $\int_{\Omega} |f(x)|^p \, dx = \infty$. - **Example**: For $f(x) = \frac{1}{x}$ on $[0, 1]$ and $p = 1$, $ \int_{0}^{1} |x^{-1}|^1 \, dx = \int_{0}^{1} \frac{1}{x} \, dx = \infty. $ Thus, $f$ is not 1-integrable on $[0, 1]$. ##### 2. Comparison with Known Divergent Integrals - **Comparison Test for Divergence**: If $|f(x)|^p \geq h(x)$ for all $x$ in a subset of $\Omega$ and $\int h(x) \, dx = \infty$, then $f$ is not $p$-integrable. - **Example**: For $f(x) = \frac{1}{x^\alpha}$ on $[0, 1]$, if $\alpha p \geq 1$, - Since $x^{-\alpha p} \geq x^{-1}$ when $\alpha p \geq 1$, - And $\int_{0}^{1} x^{-1} \, dx = \infty$, - Therefore, $f$ is not $p$-integrable on $[0, 1]$. ##### 3. Behavior at Singularities or Infinity - **Analyze Limits**: Investigate the behavior of $|f(x)|^p$ as $x$ approaches points where $f$ may become unbounded (e.g., near zero or infinity). - **Example**: If $f(x)$ grows too rapidly as $x \to \infty$, the integral may diverge. #### Techniques and Tips - **Break Up the Domain**: If the domain $\Omega$ is unbounded or $f$ has singularities, consider splitting the integral into parts where $f$ behaves differently. - **Use of Integral Tests**: Apply integral tests from calculus, similar to those used in convergence tests for series. - **Apply Hölder’s or Minkowski’s Inequalities**: These can help in estimating integrals, especially when dealing with products or sums of functions. - **Consider Transformations**: Sometimes, a substitution can simplify the integral, making it easier to evaluate. #### Summary - **To Prove $p$-Integrability**: - Show that $\int_{\Omega} |f(x)|^p \, dx$ is finite via direct computation, estimation, or comparison with a known convergent integral. - **To Prove Non-$p$-Integrability**: - Demonstrate that $\int_{\Omega} |f(x)|^p \, dx$ is infinite, often by comparing $|f(x)|^p$ to a known function whose integral diverges. #### Example Problems 1. **Determine if $f(x) = \frac{\sin x}{x}$ is 2-integrable on $(0, \infty)$:** - **Approach**: Estimate $|f(x)|^2 \leq \frac{1}{x^2}$ for large $x$. - **Conclusion**: Since $\int_{1}^{\infty} x^{-2} \, dx = 1$, which is finite, and $f$ is bounded near $x = 0$, $f \in L^2(0, \infty)$. 2. **Check if $f(x) = x^{-\frac{1}{2}}$ is 2-integrable on $[0, 1]$:** - **Compute**: $\int_{0}^{1} x^{-1} \, dx = \infty$. - **Conclusion**: $f \notin L^2(0, 1)$. #### Final Thoughts - The key to determining $p$-integrability lies in understanding the behavior of $|f(x)|^p$ over the domain $\Omega$. - Pay special attention to points where $f(x)$ may become large or unbounded. - Utilize a combination of analytical techniques, comparisons, and known integral results to make your case. By systematically analyzing the integral of $|f(x)|^p$ and employing appropriate mathematical tools, you can effectively determine whether a function is $p$-integrable over a given domain.