Question

Prove: If \(f\) is locally integrable on \([a, b),\) then \(\int_{a}^{b} f(x) d x\) exists if and only if for each \(\epsilon>0\) there is a number \(r\) in \((a, b)\) such that $$ \left|\int_{x_{1}}^{x_{2}} f(t) d t\right|<\epsilon $$ whenever \(r \leq x_{1}, x_{2}

Short answer

In conclusion, a function \(f\) is locally integrable on \([a, b)\) and \(\int_{a}^{b} f(x) dx\) exists if and only if for each \(\epsilon > 0\), there exists a number \(r\) belonging to \((a, b)\) such that the integral $$\left|\int_{x_{1}}^{x_{2}} f(t) dt\right| < \epsilon$$ whenever \(r \leq x_1, x_2<b\). We have proven this statement through both directions - assuming the existence of the integral and assuming the given condition on the integral. The proof involved using the continuity of the integral function and concepts such as uniformly convergent Riemann sums.

Step-by-step solution

Proof (\(\Rightarrow\)): Assuming \(\int_{a}^{b} f(x) dx\) exists

: First, let's assume that \(\int_{a}^{b} f(x) dx\) exists. We need to prove that for a given \(\epsilon > 0\), there exists a number \(r\) in the interval \((a, b)\) such that the integral under the given conditions is bounded by \(\epsilon\). Since \(f\) is locally integrable on \([a, b)\) and \(\int_{a}^{b} f(x) dx\) exists, this means that \(\int_{a}^{x} f(t) dt\) is continuous. So, for a given, \(\epsilon > 0\), as \(\int_{a}^{x} f(t) dt\) is continuous, there should exist a \(\delta > 0\) such that for any \(x_1, x_2 \in [a + \delta, b)\): $$\left|\int_{x_1}^{x_2} f(t) dt\right| = \left|\int_{a}^{x_2} f(t) dt - \int_{a}^{x_1} f(t) dt\right| < \epsilon$$ Now, set \(r = a + \delta\). This shows that the given condition is satisfied if \(\int_{a}^{b} f(x) dx\) exists.

Proof (\(\Leftarrow\)): Assuming the integral under given conditions is bounded by epsilon

: Now we need to prove the converse - if the given condition on the integral is satisfied, then \(\int_{a}^{b} f(x) dx\) must exist. Given that for each \(\epsilon > 0\) there is a number \(r\) in \((a, b)\) such that $$\left|\int_{x_{1}}^{x_{2}} f(t) dt\right| < \epsilon$$ whenever \(r \leq x_1, x_2<b\). We want to prove that \(\int_{a}^{b} f(x) dx\) exists. Since \(f\) is locally integrable on \([a, b)\), by applying Exercise 2.1.38, we find that for any sequence of partitions \(P_n\) with \(||P_n||\to0\), the limit of the Riemann sums converges to some value. Denote \(A = \int_{a}^{r} f(x) dx\), we have \(\lim_{n\to\infty}\sum_{i=1}^{n}f(t_i)\Delta x_i=A\). Since \(|\int_{x_1}^{x_2} f(t)dt| < \epsilon\), we also have \(|\sum_{i=r}^{x_1}f(t_i)\Delta x_i - \sum_{i=r}^{x_2}f(t_i)\Delta x_i| < \epsilon\) whenever \(r \leq x_1, x_2<b\). Therefore, the Riemann sums converge uniformly on \([r, b)\). By employing the linearity of the integral and considering the finite integral in the interval \([a, r]\), it can be concluded that the integral over the whole interval \([a, b)\) exists, i.e., \(\int_{a}^{b} f(x) dx\) exists. Thus, we've proven both directions of the statement, and the proof is complete.

Key concepts

Locally Integrable Functions

In mathematical analysis, locally integrable functions are an essential concept. A function is considered locally integrable on an interval if it can be integrated over any compact subset of that interval. This means that for any two points within the interval, the integral of the function from one point to the other is finite.

• Locally integrable does not mean the function is globally integrable; it only guarantees integrability on small, local spans of the interval.
• For example, a function that may have issues like discontinuities or vertical asymptotes can still be locally integrable because these problematic points may not affect the entire interval.
• Solutions of differential equations and certain types of mathematical series often rely on locally integrable functions.

The significance of locally integrable functions ties back to the existence of Lebesgue integrals. With this property, we can conclude that on any small segment within the interval, the function is well-behaved enough to let us perform integration. This is crucial for establishing many results in real analysis and helps us use simpler methods like Riemann integration in a piecewise manner across the interval.

Riemann Sums

Riemann sums are a fundamental concept in calculus, providing a method to approximate definite integrals. They are especially important for understanding how we calculate integrals and, by extension, areas under curves.

• A Riemann sum is essentially the sum of the products of function values at certain points and corresponding subinterval lengths of a partition.
• In mathematical terms, given a partition of an interval \[a, b)\] and a function \(f(x)\), a Riemann sum is expressed as \( \sum_{i=1}^{n} f(x_{i}^{*}) \Delta x_{i} \), where \(\Delta x_{i} = x_{i} - x_{i-1}\).
• By choosing different types of sample points \(x_{i}^{*}\) such as left-endpoints, right-endpoints, or midpoints, different forms of Riemann sums are obtained.

Riemann sums help us approximate the integral by summing up areas of rectangles under a curve, which becomes more accurate as we increase the number of subdivisions and decrease the width of each subinterval. The principle technique they offer is crucial for transitioning from a numerical approximation to an exact integral as the partitions become finer.

Convergence of Integrals

The concept of convergence of integrals is fundamental when dealing with improper integrals or when extending the notion of the integral beyond cases where the function satisfies the criteria for standard Riemann integrability.

• Convergence of an integral typically means arriving at a finite value, often as the limit of a sequence of better and better approximations.
• When dealing with Lebesgue integration, convergence is focused on ensuring the integral result remains finite even if the interval approaches infinity or a singular point.
• For a function to have a convergent integral over an interval, any oscillations or singularities within the function must be manageable, which means they must not cause the integral over infinite limits or at problematic points to diverge.

Understanding the convergence of integrals allows mathematicians and students to handle cases where traditional techniques do not apply. Through this concept, integrals are extended to work with more complex functions and domains. This is particularly crucial in theoretical pursuits and practical applications in fields like physics, engineering, and economics.