Section 5.3

Sequences of Measurable Functions

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Problem 5.3.1

Prove (a)-(e) of (MF13).

Recall the definition of the positive and negative parts of $f: A \rightarrow \overline{\mathbb {R}}$ : $f^{+}(\omega ) = \max \left\{ f(\omega ), 0\right\}$ and $f^{-}(\omega ) = -\min \left\{ f(\omega ), 0\right\}$ for every $\omega \in A$ (so $f^{+}, f^{-} \geqslant 0$ everywhere, at most one of $f^{+}(\omega ), f^{-}(\omega )$ is nonzero for each $\omega$ , and $f = f^{+}-f^{-}$ ).

(MF13): Let $f: A \rightarrow \overline{\mathbb {R}}$ , $g: A \rightarrow \overline{\mathbb {R}}$ , and $h: A \rightarrow \overline{\mathbb {R}}$ be arbitrary. Then $f$ is $\mathcal{F} / \mathfrak {B}^{*}$ -measurable iff both $f^{+}$ and $f^{-}$ are $\mathcal{F} / \mathfrak {B}^{*}$ -measurable. In addition:

(a)

$\left|f\right| = f^{+}+f^{-}$ . In particular, $f \geqslant 0$ iff $f=f^{+}$ and $f^{-}=0$ . Also, $f<0$ iff $f^{-}=-f$ and $f^{+}=0$ .

(b)

Let $c \in \mathbb {R}$ . Then $(cf)^{+} = cf^{+}$ if $c \geqslant 0$ , $-cf^{-}$ if $c<0$ ; and $(cf)^{-} = cf^{-}$ if $c \geqslant 0$ , $-cf^{+}$ if $c<0$ .

(c)

We do not necessarily have $(f+g)^{+}=f^{+}+g^{+}$ , nor do we necessarily have $(f+g)^{-}=f^{-}+g^{-}$ , but we do have $0 \leqslant (f+g)^{+} \leqslant f^{+}+g^{+}$ and $0 \leqslant (f+g)^{-} \leqslant f^{-}+g^{-}$ .

(d)

If $\left|g\right| \leqslant f$ then $0 \leqslant g^{+}, g^{-} \leqslant f$ .

(e)

If $f=g-h$ and $g, h \geqslant 0$ , then $f^{+} \leqslant g$ and $f^{-} \leqslant h$ .

Problem 5.3.2

(a)(b)

The class of $\mathcal{F} / \mathfrak {B}^{*}$ -measurable functions is not necessarily closed under uncountable suprema and infima. The following outline gives a simple instantiation of this claim. Let $\Omega =\mathbb {R}$ and $\mathcal{F}=\mathcal{B}$ .

(a)

Let $E$ denote a non-Borel set as constructed in Section 4.5. Argue that $E$ cannot be at most countable.

(b)

For each $x \in E$ , define $f_{x}: \mathbb {R} \rightarrow \overline{\mathbb {R}}$ by writing $f_{x}(\omega )=\; \mathbb {1}_{\left\{ x\right\} }(\omega )$ for each $\omega \in \mathbb {R}$ . Then $f_{x}$ is $\mathcal{F} / \mathscr {B}^{*}$ -measurable for each $x \in E$ , but $\sup_{x \in E} f_{x}=\; \mathbb {1}_{E}$ , hence $\sup_{x \in E} f_{x}$ is not $\mathcal{F} / \mathscr {B}^{*}$ -measurable.