Sums and integration against counting measure: Appendix 4: More details about liminf

For other posts in this series, see

https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure

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In this appendix, we fill in some details about claims made in Part IV concerning lim inf.

We will use the same definitions as in Part IV concerning the lim inf of nets in ¯R.

Now let J be a (non-empty) indexing set, and suppose that we have non-negative extended real numbers (aj)jJ and (bj)jJ. We claim that
inf{aj+bj:jJ}inf{aj:jJ}+inf{bj:jJ}.
This is fairly standard, but here is a quick proof for those in any doubt.

Set a=inf{aj:jJ} and b=inf{bj:jJ}. Then, for each jJ, we have aja and bjb. Thus (noting that all values are in [0,], so there is no danger when adding) we have
aj+bja+b.
The inequality (1) now follows immediately (either by the definition of infimum, or by taking the infimum over jJ).

Next suppose that (yα)αA and (zα)αA are both nets in [0,] (where A is a directed set). In Part IV of the main series we claimed that

lim infα(yα+zα)lim infαyα+lim infαzα.

To see this, for each αA, set

sα=inf{yβ:βA,βα}[0,], and

tα=inf{zβ:βA,βα}[0,].

Also set

y=lim infαyα=supαAsα=limαsα, and

z=lim infαzα=supαAtα=limαtα.

Note that, for each αA, by (1),

inf{yβ+zβ:βA,βα}inf{yβ:βA,βα}+inf{zβ:βA,βα}=sα+tα.

Taking the limit over α of both sides gives us (2).

It would be nice to avoid using the algebra of limits here by taking the supremum of both sides instead. Unfortunately the standard inequality for the supremum of sums is the wrong way round! However, the nets sα and tα are monotone increasing here, so it is actually true in this setting that

supα(sα+tα)=supαsα+supαtα.

So (if you check the details of that claim) you don't have to appeal to the algebra of limits.

Now let X be a set with exactly two elements, say X={x1,x2}, and let (fα)αA be a net of functions from X to [0,]. Taking yα=fα(x1) and zα=fα(x2), (2) gives us

lim infα(fα(x1)+fα(x2))lim infαfα(x1)+lim infαfα(x2), which we can rewrite as

xXlim infαfα(x)lim infαxXfα(x).

That gives us Fatou's Lemma for nets and sums in the case where X has two points. We'll repeat that in the next appendix, as part of our proof for general sets.


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