Ask for explanation of some notations in Ethereum opcodes

Question

I am going to implement EVM for my own project, while reading yellow paper, I can not understand some notations, can anybody help me to clarify them:

what is µ_i in the picture above

Answer 1

From the second paragraph of section 3 in the paper:

The two sets of highly structured, ‘top-level’, state values, are denoted with bold lowercase Greek letters. They fall into those of world-state, which are denoted σ (or a variant thereupon) and those of machine-state, µ.

So µ represents the state inside the EVM. µ' represents the new EVM state, after some transition has been applied. For example:

(133) µ'_s ≡ µ_s + ∆

Further in section 3:

Square brackets are used to index into and reference individual components or subsequences of sequences, e.g. µ_s[0] denotes the first item on the machine’s stack. For subsequences, ellipses are used to specify the intended range, to include elements at both limits, e.g. µ_m[0..31] denotes the first 32 items of the machine’s memory

Finally, your i query (section 222):

µ'_i is defined as this new maximum number of words of active memory;

Meaning µ_i was the number of words of active memory before any change was made in the EVM.