If $f_n$ $\longrightarrow$ $f$ in measure and $g_n$ $\longrightarrow$ $g$ in measure then show that $f_n +g_n$ $\longrightarrow$ $f+g$ in measure

Question

(1) If $f_n$ $\longrightarrow$ $f$ in measure and $g_n$ $\longrightarrow$ $g$ in measure then show that $f_n +g_n$ $\longrightarrow$ $f+g$ in measure

(2)If $f_n \longrightarrow \theta$ in measure and $g_n \longrightarrow \theta$ in measure where $\theta$ is the identically zero function on $\Omega$ then show that $f_ng_n \longrightarrow 0$

My attempt For both cases I use the fact that $(\Omega,\Sigma,\mu)$ is my measure space where everything is defined in the H.L Royden.

For (1) it follows from the relation $ {x \in E |(f_n+g_n)(x)-(f+g)(x)}|\geq \epsilon$}

the above is a subset of the set $|f_n(x)-f(x)|\geq \epsilon/2$ obviously and thus is the

union of the set $|g_n(x)-g(x)|\geq \epsilon/2$ and hence the resukt holds.

Note here my measure is taken over an arbitrary set E. And this shows (1)

For (2) im drawing a blank and I don't have any idea how to even begin such a question. How do I 'handle' the zero function ?

Answer 1

Hints for (2):

1. Let $(h_n)_{n \in \mathbb{N}}$ be a sequence of measurable functions. Prove that $h_n \to 0$ in measure implies $h_n^2 \to 0$ in measure.
2. Use the first part of your problem and Step 1 to show that $(f_n+g_n)^2 \to 0$ in measure and $(f_n-g_n)^2 \to 0$ in measure.
3. Use $$f_n \cdot g_n = \frac{1}{4} \big( (f_n+g_n)^2- (f_n-g_n)^2 \big)$$ and Step 1 to conclude that $f_n \cdot g_n \to 0$ in measure.