Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a non-negative continuous function, and let $$ \mu([a, b)):=\int_a^b f(x) \mathrm{d} x, \quad-\infty<a \leqslant b<+\infty . $$ Prove that this formula defines a Lebesgue-Stieltjes measure on the real line. ( $f$ is called the density of such a measure.) Write $F$.

   Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a non-negative continuous function, and let
$$
\mu([a, b)):=\int_a^b f(x) \mathrm{d} x, \quad-\infty<a \leqslant b<+\infty .
$$

Prove that this formula defines a Lebesgue-Stieltjes measure on the real line. ( $f$ is called the density of such a measure.) Write $F$.

Show more…

Integral and Measure

Integral and Measure

Vigirdas Mackevicius 1st Edition

Chapter 15, Problem 5

Instant Answer

Step 1

Since $f$ is non-negative, the integral of $f$ over any interval is also non-negative. Show more…

Show all steps

lock

Thanks for your feedback!

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a non-negative continuous function, and let $$ \mu([a, b)):=\int_a^b f(x) \mathrm{d} x, \quad-\infty<a \leqslant b<+\infty . $$ Prove that this formula defines a Lebesgue-Stieltjes measure on the real line. ( $f$ is called the density of such a measure.) Write $F$.

Play audio

Feedback

Powered by NumerAI

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later