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Analysis Integral confused concept. Please clear writing and follow the comment 1. upper sum with respect to...

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Advance Math

Analysis Integral confused concept. Please clear writingand follow the comment

1. upper sum with respect to P is U(f,P)= sum ofMk(xk-xk-1).Does that equal to U(f)=inf{u(f,P): p is a element of Q} (Q is thecollection of all possible partition})

Also, is Mk=sup{f(x):x is a element of{xk-1,xk}?????

2. If Q is a refinement of P, then U(f, Q)<=U(f, P) and L(f,P),<=L(f,Q). I don't understand the geometric meaning behindthis because Q has more element than P and contains all of theelements from P, However, why U(f, Q)<=U(f, P)????

This Q is not the Q from question1

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