For each \(n ∈ \mathbb{N} \text {, let } f_n:[0,1] \rightarrow \mathbb{R}\) be a measurable function such that \(\left|f_n(t)\right| \leq \frac{1}{\sqrt{t}}\) for all t ∈ (0, 1). Let \(f:[0,1] \rightarrow \mathbb{R}\) be defined by f(t) = 1 if t is irrational and f(t) = -1 if t is rational. Assume that fn(t) → (t) as n → ∞ for all t ∈ [0, 1]. Then
1
f is not measurable
2
f is measurable and \(\int_{[0,1]} f_n d \mu \rightarrow 1 \text { as } n \rightarrow \infty\)
3
f is measurable and \(\int_{[0,1]} f_n d \mu \rightarrow 1 \text { as } n \rightarrow \infty\)
4
f is measurable and \(\int_{[0,1]} f_n d \mu \rightarrow-1 \text { as } n \rightarrow-\infty\)
5
Question Not Attempted