Let \(f:[0,1]\to[0,\infty)\) be lower semicontinuous. For a partition \[ P:0=t_0<t_1<\cdots<t_N=1, \] define the lower Riemann sum \[ S_+(f,P)=\sum_{j=1}^N\left(\inf_{x\in[t_{j-1},t_j)}f(x)\right)(t_j-t_{j-1}). \] Let the mesh be \[ |P|=\max_j(t_j-t_{j-1}). \] Prove that \[ S_+(f,P)\longrightarrow\int_0^1 f(x)\,dx \] as \(|P|\to0\), where the integral is the Lebesgue integral.
Solution
For a partition \(P\), define the lower-step function \[ s_P(x)=\inf_{y\in[t_{j-1},t_j)}f(y) \qquad\text{when }x\in[t_{j-1},t_j). \] Then \[ S_+(f,P)=\int_0^1 s_P(x)\,dx \] and \[ 0\le s_P(x)\le f(x) \] for \(x<1\).
Now let \((P_k)\) be any sequence of partitions with \(|P_k|\to0\). Fix \(x\in[0,1)\). Since \(f\) is lower semicontinuous, for every \(\varepsilon>0\) there exists \(\delta>0\) such that \[ |y-x|<\delta\implies f(y)>f(x)-\varepsilon. \] For all sufficiently large \(k\), the subinterval of \(P_k\) containing \(x\) has length less than \(\delta\), hence is contained in \((x-\delta,x+\delta)\). Therefore \[ f(x)-\varepsilon\le s_{P_k}(x)\le f(x). \] Thus \[ s_{P_k}(x)\longrightarrow f(x) \] for every \(x\in[0,1)\).
By Fatou’s lemma, \[ \int_0^1f\le\liminf_{k\to\infty}\int_0^1s_{P_k}. \] On the other hand, \(s_{P_k}\le f\), so \[ \limsup_{k\to\infty}\int_0^1s_{P_k}\le\int_0^1f. \] Hence \[ S_+(f,P_k)=\int_0^1s_{P_k}\longrightarrow\int_0^1f. \] Since this holds for every sequence with mesh tending to zero, the desired limit follows.