Signed measure inequality |v1+v2|<=|v1|+|v2|

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Homework Statement

The problem is Excercise 5. in page 88 of Folland's "real analysis: modern techniques and their applications", 2nd edition, as the image below shows.

Homework Equations

As the hint indicates, we should use Excercise 4.

The Attempt at a Solution

From Excercise 4, if signed measure $$\nu=\lambda-\mu$$, then $$|\nu|=\nu^++\nu^-\leq\lambda+\mu$$. Supposing $$\nu_1=\lambda_1-\mu_1$$, $$\nu_2=\lambda_2-\mu_2$$, then $$|\nu_1+\nu_2|=(\nu_1+\nu_2)^++(\nu_1+\nu_2)^-$$, If we can prove $$(\nu_1+\nu_2)^+\leq\nu_1^++\nu_1^-=|\nu_1|$$ and $$(\nu_1+\nu_2)^-\leq\nu_2^++\nu_2^-=|\nu_2|$$, then the result is obtained. But I can not prove these inequality, please help me, thanks!

 

[mighty2000]

Thank you for bringing this problem to our attention. After reviewing the exercise and the hint provided, I believe I can offer some guidance on how to approach this problem.

Firstly, let's look at the given hint. It suggests using Exercise 4, which deals with the absolute value of a signed measure. Remember that the absolute value of a signed measure, denoted by |ν|, is defined as |ν| = ν^+ + ν^-, where ν^+ and ν^- are the positive and negative variations of ν, respectively.

Now, let's consider the two signed measures given in the problem, ν_1 = λ_1 - μ_1 and ν_2 = λ_2 - μ_2. We want to show that |ν_1 + ν_2| ≤ |ν_1| + |ν_2|. Using the definition of the absolute value of a signed measure, we have |ν_1 + ν_2| = (ν_1 + ν_2)^+ + (ν_1 + ν_2)^-.

To simplify this expression, we can use the properties of signed measures and the fact that ν_1 and ν_2 are both signed measures. For example, we know that (ν_1 + ν_2)^+ = ν_1^+ + ν_2^+ and (ν_1 + ν_2)^- = ν_1^- + ν_2^-. This is because the positive and negative variations of a signed measure are additive.

Now, using these properties, we can rewrite the expression as (ν_1 + ν_2)^+ + (ν_1 + ν_2)^- = (ν_1^+ + ν_2^+) + (ν_1^- + ν_2^-).

Next, we can use the given hint from Exercise 4, which states that if ν = λ - μ, then |ν| ≤ λ + μ. In our case, we have ν_1 = λ_1 - μ_1 and ν_2 = λ_2 - μ_2. So, using the hint, we can write |ν_1| = |λ_1 - μ_1| ≤ λ_1 + μ_1 and |ν_2| = |λ_2 - μ_2| ≤ λ_2 + μ_2