# Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions

Show that lower semicontinuous function $f:X\rightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions.

My attempt: I don't know how to approximate $f(x)$ to within $[f(x)-1/n,f(x)]$ by $h_n(x)$ which is a linear combination of characteristic function of open sets, using lower semicontinuity of $f$.

Can anyone help?

• Now sure how to proceed with your idea. But from a post a couple years back... Let $f_k(x)=\inf\{f(y)+kd(x,y):y\in X\}.$ – matt biesecker May 13 '15 at 5:00

The function $$f(x)$$ may be approximated to within $$[f(x)-1/n, f(x)]$$ by the function $$h_{n} = \frac{1}{n}\sum_{k = 1}^{n-1}\chi_{f^{-1}\left(\left(\frac{k}{n}, 1\right]\right)}$$ where $$\chi_{A}$$ denotes the characteristic function of a subset $$A$$ of $$X$$. Note that $$h_{n}$$ is a linear combination of characteristic functions of open sets.
The characteristic function of any open set $$A$$ in a metric space $$(X, \rho)$$ can be written as the supremum of an increasing sequence of continuous functions. To prove this, note that every open set in a metric space is $$F_{\sigma}$$. In particular, if $$A$$ is an open set then $$A$$ can be written as the union of an increasing sequence $$A_{1}\subseteq A_{2}\subseteq\cdots$$ of closed sets where $$A_{n} = \left\{y\in X: \rho(y, X-A)\geq \frac{1}{n}\right\}.$$ For each positive integer $$n$$ define $$g_{n}:X\rightarrow\mathbb{R}$$ by $$g_{n}(x) = \min\{n\rho(x, X-A), 1\}$$ and verify that $$\chi_{A} = \sup_{n}g_{n}$$.
For each $$1\leq k\leq n-1$$ let $$\frac{1}{n}\chi_{f^{-1}\left(\left(\frac{k}{n}, 1\right]\right)} = \sup_{m}g_{k,m}$$ for an increasing sequence of continuous functions $$\{g_{k, m}\}_{m\in\mathbb{Z}^{+}}$$. One can verify that $$h_{n} = \sup_{m} h_{n, m}$$ where $$h_{n, m} = \sum_{k = 1}^{n-1}g_{k,m}$$.
Finally, one can show that $$f = \sup_{(n, m)\in\mathbb{Z}^{+}\times\mathbb{Z}^{+}}h_{n, m}.$$ Using any bijection between $$\mathbb{Z}^{+}\times\mathbb{Z}^{+}$$ and $$\mathbb{Z}^{+}$$, rearrange the functions $$\{h_{n,m}\}$$ into a sequence $$\{\tilde{k}_{n}\}$$ and let $$k_{n} = \max\{\tilde{k}_{1},\dots,\tilde{k}_{n}\}$$. Then $$\{k_{n}\}$$ is the required sequence of increasing continuous functions such that $$f = \sup_{n}k_{n}$$.