Counterexample for this statement about the Riemann integral?

Question

I'm considering a Riemann integral, and trying to work out if the following statement is true: Let $P$ and $P'$ be two partitions such that $\mu (P') \lt \mu (P)$. $\mu(P)$ denotes the mesh of P, the length of the largest interval in the partition). Then, $$U(f,P')\le U(f,P) \quad L(f,P')\ge L(f,P)$$ For any bounded function $f$ from an interval to the reals. This is such a general statement and seemingly useful one that I would be surprised that it is true because we were never taught this. But, I can't seem to come up with a counterexample. Could someone help?

Answer 1

It's false. On the interval $[0,1]$, consider the function $f = 1_{[0, 0.1]}$ which takes the value 1 on the interval $[0,0.1]$ and $0$ otherwise. Let $P = \{0, 0.2, 1\}$ and $P' = \{0, 0.5, 1\}$. Then $\mu(P) = 0.8 > 0.5 = \mu(P')$, however $U(f,P) = 0.2$ and $U(f, P') = 0.5$.

The problem is that even though $P$ has at least one large subinterval, it might be in a region where the function doesn't vary much, so it may not actually work against the ability of $U(f,P)$ to be a good approximation to the integral.