For a few weeks I've been working on this neat copy of my own attempt to prove some things about step functions. The image links below take you to snapshots of what is around two pages worth of lemmas and one small theorem at the end. I've included the links in order from top to bottom, but they overlap just in case. I have tried to be very precise and as such, appreciate any time anyone takes to give this stuff a look:

https://gyazo.com/d84aea07bc005c9817a1bc69098ef155

https://gyazo.com/72f6a772ec1c76202c0b604f8edbdff4

https://gyazo.com/97e29161ed025d4813d1ec79dea5930b

https://gyazo.com/94abc31cda51493b66a30368630730f3

Regards,

Magnitude.