# Mapping reduction from $A_{TM}$ to $INFINITE_{TM}$ same as to $ALL_{TM}$?

I was trying to solve a problem with a mapping reduction from $$A_{TM}$$ to $$INFINITE_{TM}$$, and came across a solution that was 100% identical to another solution I saw for $$A_{TM} \leq_M ALL_{TM}$$. This is the reduction:

$$M_f$$: Given an input of $$\langle M, w\rangle$$, where $$M$$ is a $$TM$$ and $$w$$ is a word:

Define a TM $$M_1$$:

Given the input $$x$$ do:

1. Run $$M$$ on $$w$$ and return its result
2. Return $$\langle M_1\rangle$$

This is correct for $$ALL_{TM}$$ because if $$M$$ accepts $$w$$, $$L(M_1)=\Sigma^*$$, whereas if it rejects, $$L(M_1)=\emptyset$$

However, it is also given as a solution for $$A_{TM} \leq_M INFINITE_{TM}$$, where $$INFINITE_{TM}$$=$$\{\langle {M \rangle} |$$ $$M$$ is a $$TM$$, $$L(M)$$ is an infinite language $$\}$$, for example here: http://www.sfu.ca/~kabanets/308/lectures/lec7.pdf

I don’t fully understand why theirs, or other similar explanations, are correct. What about infinite languages that aren’t $$\Sigma^*$$? What about finite languages that aren’t $$\emptyset$$?