Two kinds of Metrics of Convergence in Measure

Question

There are many kinds of metrics that can induce the topology of convergences in measure. Two most common metrics are

1. Here is the first one.

$ d(f,g) := \inf_{\delta > 0} \big(\mu(|f-g|>\delta) + \delta\big) $

2. Here is the second one, which is the most common.

$ d(f,g) = \int \frac{|f-g|}{1+ |f-g|}d\mu $

I have three questions:

Q1 I am trying to prove the Triangle inequality of the first metric. I have done so far:

$ \mu\left\{ x\in X:\left|f\left(x\right)-g\left(x\right)\right|>\delta\right\} +\delta \le \mu\left\{ x\in X:\left|f\left(x\right)-h\left(x\right)\right|>\frac{\delta}{2}\right\} +\frac{\delta}{2}+\mu\left\{ x\in X:\left|h\left(x\right)-g\left(x\right)\right|>\frac{\delta}{2}\right\} +\frac{\delta}{2} $

But I have no idea what to do next.

Q2 What is the difference between the two metrics? Both metrics can induce the topology of convergence in measure. As far as I am concerned, the second metric is used only in probability contexts or in the case the measure is finite. I wonder whether this statement is right. If it is true, what's wrong with the second metric when the measure is not finite?

Q3 Given a measure space $\left(X,\mathscr{F},\mu\right)$ and let $L^{0}\left(X,\mathscr{F},\mu\right)$ be the vector space of all real-valued measurable functions on $\left(X,\mathscr{F},\mu\right)$. If both metrics can be defined on $L^{0}\left(X,\mathscr{F},\mu\right)$, are the topologies of these two spaces the same?

Can anyone help me out? Thanks in advance.

Answer 1

Q1: let $A=d(f,h)$ and $B=d(h,g)$. Let $\epsilon >0$. Then there exist $\delta_1 >0$ and $\delta_2 >0$ such that $\mu(|f-h| >\delta_1)+\delta_1 <A+\epsilon$ and $\mu(|h-g| >\delta_2)+\delta_2 <B+\epsilon$. Note that $|f-g| >\delta_1+\delta_2$ implies that either $|f-h| >\delta_1$ or $|h-g| >\delta_2$. Hence $d(f,g) \leq \mu(|f-g| >\delta_1+\delta_2\leq A+B+2\epsilon$. Since $\epsilon$ is arbitrary we get $d(f,g) \leq d(f,h)+d(h,g)$.

If $\mu$ is not finite it is not true that all bounded measurable functions are integrable. The second definition cannot be used for infinite measures since metrics cannot take the value $\infty$.

For a finite measure both metrics yield the same topology since $f_n \to f$ in one metric iff $f_n \to f$ in the other.