sudden thoughts, upon waking .. [added: and before sleeping ]

upon opening my eyes this morning, i thought immediately of two things:

1. you know, it's cold in this apartment.
   did i turn the heat on, last night?
   
   
2. wait. if our metric space is the entire plane, then ..
   <insert unspecified singu1ar mea$ure here>
   
   .. shouldn't be doub1ing. so does it mean that ..
   <insert crazy idea here>
   
   .. will work?!?

mind you, this came to my mind without caffeine;
amazing!

as it is a teaching day today, i haven't had time to work out the details. subsequently, today i don't like teaching very much.

---

i had time to think about the details, after all. it's still not clear whether the crazy idea will work. on the bright side, is that it reduces to invoking the theorems of others and reading their work carefully.

on a related note,

1. sometimes i wish that a paper of a1berti, cs0rnyei, and prei$s were ready; it's the work in preparation that concerns sets on which Lips¢hitz functions are n0ndifferenτiab1e. at this point i really do want to know some details about how they do what they say that they do.
   
   
2. i know many mathematicians with very clear motivations. they are specialists, experts in their field and from careful, deep analysis, they prove good results. several words come to mind:
   
   centered,
   focused.
   
   i have no such qualities. maybe it is a happenstance of youth and having just started an independent, mathematical life, but it is hard to focus on one single subject and then to work deliberately.
   
   there are just so many interesting things to study. as a result, i seem to recall quite a few facts, but know how to prove very few things.
   
   the few theorems i have in mind have chimeric proofs; they are often formed from borrowing random ideas from different theories. the arguments would be valid enough, but for step 1 i need this theory and for step 2 i'll have to introduce notation and lemmas from another theory.
   
   a proof is a proof, but a good proof should explain itself. a good proof should adhere to morals. put that way, my proofs are immoral scrawlings!