Product of two integrable function is integrable with respect to product measure

Question

I know this question has been answered before, but it has not answered my confusion.

The question is like this:

$f$ and $g$ are integrable, show that $h(x,y)=f(x)g(y)$ is integrable with respect to product measure then $\int f(x)g(y) d(\mu\times \nu)=\int f(x)(\int g(y)d\nu)d\mu$. And a hint for this problem is to use Fubini, and $\mu,v$ is not necessarily sigma finite.

I am thinking that in order to use Fubini's theorem, $h$ must be integrable. I can show that this is true by building up from characteristic functions, but it will get $\int f(x)g(y) d(\mu\times \nu)=\int f(x)(\int g(y)d\nu)d\mu$ along the process. So this the hint true or ???

Answer 1

The other answer and the comments therein do not actually answer this question. Also Fubini's Theorem does not assume $ \sigma $-finiteness, but rather completeness, of the measure spaces. Let's say we're working with the measure spaces $ (X, \mathfrak{A}, \mu) $ and $ (Y, \mathfrak{B}, \nu) $. The hypothesis, then, is that $ \mu $ and $ \nu $ are complete measures (I'd assume), and that $ f \in L^1(X, \mathfrak{A}, \mu) $ and $ g \in L^1(Y, \mathfrak{B}, \nu) $.

We want to show that $ h(x, y) := f(x)g(y) $ is $ \mu \times \nu $-integrable, and that

\begin{equation*} \int h(x, y) d(\mu \times \nu)(x, y) = \int f(x) d\mu(x) \cdot \int g(y) d\nu(y) \end{equation*}

This can done by painful bootstrapping (though there is probably a simpler procedure):

1. Suppose $ f(x) $ and $ g(y) $ are simple functions, say, $ f(x) = \sum\limits_{m=1}^{M} a_n \chi_{A_m}(x) $ and $ g(y) = \sum\limits_{n=1}^{N} b_n \chi_{B_n}(y) $ in canonical form (i.e., each $ A_m $ is the preimage of $ \{ a_m \} $, and each $ B_n $ is the preimage of $ \{ b_n \} $). Then

\begin{align*} \int f(x)g(y) d(\mu \times \nu)(x, y) &= \int \left( \sum\limits_{m=1}^{M} a_m \chi_{A_m}(x) \right) \cdot \left( \sum\limits_{n=1}^{N} b_n \chi_{B_n}(y) \right) d(\mu \times \nu) (x, y) \\ &= \sum\limits_{m=1}^{N} \sum\limits_{n=1}^{N} a_m \cdot b_n \cdot \int \left( \chi_{A_m}(x) \chi_{B_n}(y) \right) d(\mu \times \nu) (x, y) \end{align*}

By the definition of a simple function (in canonical form), $ A_m $ and $ B_n $ are measurable for all $ 1 \le n \le N $ and $ 1 \le m \le M $. So,

\begin{align*} \int\chi_{A_m}(x) \chi_{B_n}(y) d(\mu \times \nu)(x, y) &= \int \chi_{A_m \times B_n} (x, y) d(\mu \times \nu)(x, y) \\ &= (\mu \times \nu) (A_m \times B_n) \\ &= \mu(A_m) \nu(B_n) \end{align*}

which must be finite since $ A_m $ and $ B_n $ are measurable. $ \int fg d(\mu \times \nu) $ is, as shown above, a linear combination of these finite values, so $ fg $ is $ \mu \times \nu $-integrable. Also you can rearrange the sum:

\begin{align*} \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} a_m \cdot b_n \cdot \int \left( \chi_{A_m}(x) \chi_{B_n}(y) \right) d(\mu \times \nu) (x, y) &= \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} a_m \cdot b_n \cdot \mu(A_m) \nu(B_n) \\ &= \left( \sum_{m=1}^{M} a_m \mu(A_m) \right) \cdot \left( \sum_{n=1}^{N} b_n \nu(B_n) \right) \\ &= \left( \sum_{m=1}^{M} a_m \int \chi_{A_m} (x) d\mu(x) \right) \cdot \left( \sum_{n=1}^{N} b_n \int \chi_{B_n}(y) d\nu(y) \right) \\ &= \left( \int \left[ \sum\limits_{m=1}^{M} a_m \chi_{A_m}(x) \right] \right) \cdot \left( \int \left[ \sum\limits_{n=1}^{N} b_n \chi_{A_n}(x) \right] \right) \\ &= \left( \int f(x) d\mu(x) \right) \cdot \left( \int g(y) d\nu(y) \right) \end{align*}

So, long story short, $ \int fg d(\mu \times \nu) = \int f d\mu \cdot \int g d\nu $.

2. Suppose $ f(x) $ is $ \mu $-integrable and $ g(x) $ is $ \nu $-integrable, and both are nonnegative. Then there exist two increasing sequences of simple functions, $ \{ F_m \}_{m=1}^{\infty} $ and $ \{ G_n \}_{n=1}^{\infty} $, such that $ F_m \to f $ almost everywhere [$ \mu $] and $ G_n \to g $ almost everywhere [$ \nu $]. (I honestly don't remember how to prove that.) Therefore, $ F_n(x) G_n(y) \to f(x)g(y) $ almost everywhere [$ \mu \times \nu $]. By Case 1 and the Monotone Convergence Theorem, $ f(x)g(y) \in L^1 (\mu \times \nu) $ and

\begin{align*} \int f(x)g(y) d\mu(\mu \times \nu)(x, y) &= \lim\limits_{n \to \infty} \int F_n(x) G_n(y) d(\mu \times \nu)(x, y) \\ &= \lim\limits_{n \to \infty} \left( \int F_n(x) d\mu(x) \cdot \int G_n(y) d\nu(y) \right) \\ &= \left( \lim\limits_{n \to \infty} \int F_n(x) d\mu(x) \right) \cdot \left( \lim\limits_{n \to \infty} \int G_n(y) d\nu(y) \right) \\ &= \left( \int f(x) d\mu(x) \right) \cdot \left( \int g(x) d\nu(x) \right) \end{align*}

(Apply the MCT to $ f $, $ g $ and $ fg $.)

3. Finally, let's return to the original hypothesis that $ f $ and $ g $ are merely integrable. We can now split $ h(x, y) = f(x) g(y) $ into its negative and positive parts: $ h(x, y) = h^+(x, y) - h^-(x, y) $. We can rewrite these as

\begin{align*} h^+(x, y) = f^+(x) g^+(y) + f^-(x) g^-(y) \\ h^-(x, y) = f^+(x) g^-(y) + f^-(x) g^+(y) \end{align*}

Case 2 applies to the functions $ f^+ f^+ $, $ f^+ g^- $, $ f^- g^+ $ and $ f^- g^- $. So now (omitting the domain variables for some hope of readability):

\begin{align*} \int h \;d(\mu \times \nu) &= \int h^+ d(\mu \times \nu) - \int h^- d(\mu \times \nu) \\ &= \int \left(f^+ g^+ + f^- g^-\right)d(\mu \times \nu) - \int \left(f^+ g^- + f^- g^+ \right) d(\mu \times \nu) \\ &= \int f^+ d\mu \int g^+ d\nu + \int f^- d\mu \int g^- d\nu - \int f^+ d\mu \int g^- d\mu - \int f^- d\mu \int g^+ d\nu \\ &= \left(\int f^+ d\mu - \int f^- d\mu \right) \cdot \left( \int g^+ d\nu - \int g^- d\nu \right) \\ &= \int f d\mu \int g d\nu \end{align*}

Answer 2

Fubini's theorem states that if one of the following integrals

$$\int \left(\int |h(x,y)| \, d\mu(x)\right) d\nu(y) \quad \int \left(\int |h(x,y)| \, d\nu(y) \right) d\mu(x) \quad \iint |h(x,y)| \, d(\mu \otimes \nu)(x,y)$$

is finite, then all of these integrals are finite and

$$\int \left( \int h(x,y) \, d\mu(x) \right) d\mu(y) = \int \left( \int h(x,y) \, d\nu(y) \right) d\mu(x) = \iint h(x,y) \, d(\mu \otimes \nu)(x,y).$$

In particular, if $h(x,y) = f(x) \cdot g(y)$ with integrable $f$, $g$, then by the linearity of the integral

$$\int |h(x,y)| \, d\mu(x) = |g(y)| \left( \int |f| \, d\mu \right)$$

and therefore

$$\int \left(\int |h(x,y)| \, d\mu(x) \right) d\nu(y) = \left( \int |f| \, d\mu \right) \left( \int |g| \, d\nu \right) < \infty.$$

This means that we may apply Fubini's theorem to conclude that we can interchange the integrals. Note that this holds only true if $\mu$ and $\nu$ are $\sigma$-finite measures.